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;'-:'•
.THE
MATHEMATICAL PRINCIPLES
OF
NATURAL PHILOSOPHY.
BY
SIR ISAAC NEWTON.
Translated into English
BY ANDREW MOTTE.
TO WHICH ARE ADDED,
Newton's Syjlem of the World;
A SHORT
Comment on, and Defence of, the Principia,
BY W. EMERSON.
WITH
THE LAWS OF THE MOON'S MOTION
According to Gravity.
BY JOHN MACHIN,
Astron., Prof, at Gresh., and Sec. to the Roy. Soc.
% mto €tiitum,
(With the LIFE of the AUTHOR ; and a PORTRAIT, taken Jrom the Bud in
the Royal Obfervatory at Greenwich)
CAREFULLY REVISED AND CORRECTED BY
W. DAVIS,
Author of the ** Treatise on Land Surveying," the " Use of the Globes,"
Editor of the ^' Mathematical Companion,** &c, &c. ^c.
IN THREE VOLUMES.
VOL. II.
]Loniiont
PRINTED FOR H. D. SYMONDS, NO. 20, PATERNOSTER ROW.
%
1803.
Priiit«d by Knight & Compton, Middle Street, Cloth Fail.
» * ■
THE
MATfflEMATlCAJL PBINCIPMES
OP
NATURAL PHILOSOPHY.
OF THE MOTION OF BODIES.
BOOK 11.
SECTION I.
Of the motion of bodies that are rejifted in the ratio of the
velocity.
PROPOSITION I. THEOREM I.
If a body is rejifted in the ratio of its velocity, the motion loft
by refiftance is as thefpace gone over in its motion.
•Tor fince the motion loft in each equal particle of
time is as the velocity^ that is^ as the particle of fpace gone
oveo tben^ by composition^ the motion loft in the whole time
will be as the whole fpace gone over. Q.E.D.
CoR. Therefore if the body, deftitnte of all gravity, move
by its innate force only in free fpaces, and there be given
both its whole motion at the beginning, and alfo the motion
remaining after fome part of the way is gone over, there will
be given alfo the whole fpace which the body can defcribe in
an infinite time. For that fpace will be to the fpace now de-
fcribed as the whole motion at the beginning is to the part loft^
of that motion.
Vol. II. 3
fi MATHEMATICAL PRINCIPLES Book IL
LEMMA I.
Quantities proportional to their d^ercnces arc continually pro*
portionaL
LetAbetoA — Bas BtoB — CandCioC — 1>, fcc.
and^ by converfion^ A will be to B as B to C and C to D,
&c. Q.E.D.
PROPOSITION IL THEOREM IL
If a body-isrefifted in th& fiUio of its velocity , and moves, by
4is vis infita only ,» through afimilar medium, and the times
be taken equal, the velocities in the beginning of each of the
times are in a geometrical progrejfion, and thefpaces de^
fcribed in each of the times are as the velocities. (PI. 1,
Fig. 1.)
Case I. Let the time be divided into equal particles ; and
if at the very beginning of each particle we fuppofe the re-
iiflance to a6); with one fingle impulfe which is as the velo-
city^ the decrement of the velocity in each of the particles
of time will be as the fame velocity. Therefore the velocities
are proportional to their differences^ and therefore (by lem.
1^ book £) continually proportional. Therefore if out of an
equal number of particles there be compounded any equal por-
tions of time^ the velocities at the beginning of thofe times
will be as terms in a continued progreffion> which are taken
by intervajsj omitting every where an equal number of inter-
mediate terms. But the ratios of thefe terms are compounded
of the equal ratios of Uie intermediate terms equally repeated^
and therefore are equal. Therefore the velocities, being pro*
portional to thofe tenn% are id geometrical progreffion. Let
thofe equal particles of time be dkniiiifhed, and their number
increafed in infinitum^ fo thai the impcdfe of tefiftaaice may
become continual ; and the velooities at the beginnings of
eqml times, alwaya cotttinaally proportional, willbealfoin
this cafe continually proportional. Q.B.D*
Gass 9« And, by divifioD, the difiereneea of the vdocities,
that is^ the parts oS the vdocities loft in each of the timesi
are as the wholes; but. the fpaces defcribed in each, of the
times are as the loft parts of the velocities (by propu 1, bbofc
£), and therefore are alfo as the wholes. Q.E.D. ,
Se8. I. OP NATfXTRAL PHILOSOPHY. 3
CoBOL. Hence if to the redlangular afymptotes AC, CH>
the hyperbola B6 is defcribed, and AB, DG be drawn per-
pendicular to the aiymptote AC, and both the velocity of the
body, and the reliftance of the medium, at the very begin-
ning of the motion, be exprefled by any given line AC, and,
after fome time is elapfed, by the indefinite line DC ; the
time may be exprefled by the area ABGD, and the fpace
defcribei in that time by the line AD. For if that area, by
the motion of the point D, be uniformly increafed in the
fame manner as the time, the right line DC will decreafe in a
geometrical ratio in the fame manner as the velocity; and the
parts of the right line AC, defcribed in equal times, will de-
creafe in the fame ratio. ;
PROPOSITION III. PROBLEM I.
To define the motion of a body which, in afimilar medium^
af vends or defcends in a right line, and is refifted in the ratio
of its velocity, arid a6ted apon by an uniform force ofgra^
vity. (PL 1, Fig. 2.)
The body afcending, let the gravity be expounded by any
given re6langle BACH; and the refiftance of the medium, at
the beginning of the afcent, by the re6);angle BAD£, taken
on the contrary fide of the right line AB. Through the point
B, with die refiangular afymptotes AC, CH, defcribe an hy-
perbola, cutting the perpendiculars D£, de, in G, g; and
the body afcending will in the time DGgd defcribe the fpace
EGge ; in the time DGBA, the fpace of the whole afcent
EGB ; in the time ABKI, the fpace of defcent BFK; and in
the time iKki the fpace of defcent KFf k ; and the velocities
of the bodies (proportional to tb6 refiftance of the medium)
in thefe periods of time will be ABED, ABed, 0, ABFI,
ABfi r<efpe6Uvely ; and the greatefi: velocity which the body
can acquire by defcendihg will be BACH.
For let the reftangle BACH (PL 1, Fig. 3) be refolved into
innumerable rectangles Ak, KI, Lm, Mn, &c. which ihall
be as the increments of the velocities produced in fo many
equal times; then will 0, Ak, Al, Am, An, 8cc. be as the
whole velocities, and therefore (by fuppofition) as the refift-
ances of the medium in the beginning of each of the equal
B 2
4 MATHEMATICAL PRINCIPLES Book IL
timeg. Make AC to AK, or ABHC to ABkK, a» tlio force of
gravity to the refillance in the beginning of the fecoiul lime ;
then from the force of gravity fubdudl the refidiinces^ and
ABHC, KkHC, LIHC, MmHC, &c. will be as the alifolule
forces with which the body is a6led upon in the I>egiijning of
eachof ibe times, and therefore (by law Q) as the increments
of the velocities, that is, as the redangles Ak, Kl, Lm, Mn,
8cc. and therefore (by lem. 1, book 2) in a geometrical pro-
greffion. Therefore, if the right lines Kk, LI, Mm, Nn, &c.
are produced fo as to meet the hyperbola in q, r, s, t, &c. the
areas ABqK, KqrL, LrsM, MstN, &c. will be equal, and
therefore analogous to the equal times and equal gravitating
forces. But the area ABqK (by corol. 3, lem. 7 and 8, book
1) is to the area Bkq as Kq to |-kq, or AC to j-AK, that is,
as the force of gravity to the refiftance in the middle of the
firft time. And by the like reafoning the areas qKLr, rLMs,
sMNt, 8cc, are to the areas qklr, rims, smnt, &c. as the gra-
vitating forces to the refiftances in the middle of the fecond,
third, fourth time, and fo on. . Therefore fince the equal areas
BAKq, qKLr, rLMs, sMNt, &c. are analogous to the gra-
vitating forces, the areas Bkq, qklr, rims, smnt, &c. will be
analogous to the refiftances in the middle of each of the times,
that is (by fuppofition), to the velocities, and fo to the fpaces
defcribed. Take the fums of the analogous quantities, and
the areas Bkq, Blr, Bms, Bnt, &c. will be analogous to the
whole fpaces defcribed; and alfo the areas ABqK, ABrL,
ABsM, ABtN, &c. to the times. Therefore the body, in de-
fcending, will in any time ABrL defcribe the fpace Blr, and
in the time LrtN the fpace rlnt. Q.E.D. . And the like de-
monftration holds in afcending motion.
CoRDL. 1. Therefore the greateft velocity that the body
can acquire by falling is to the velocity acquired in any given
time as the given force of gravity which perpetually a&s
upon it to the refifting force which oppofes it at the end
of that time.
CoRo^. 2. But the time being augmented in an arithme*
ileal progreffion, the fum of that greateft velocity and the
■ *
SeS. I. Of NATURAL PHILOSOPJHY. 5
velocity in the afcent, and alfo their difference in the defq^nt^
decreaies in a geoinetrical p rogreffion.
CoKOL. 3. Alfo thie differences of the fpaces, which are
defcribed in equal differences of - the times^ decreafe in the
fame geometrical progreffion.
CoBOL. 4. The fpace defcribed by the body is the difference
of two fpaces, whereof one is as the time taken from the be*
ginning of the defcent, and the other as the velocity ; whicli
[fpaces] alfo at the beginning of the defcent are equal among
themfelves,
PROPOSITION ly. PROBLEM II'.
Suppofing the force of gravity in any Jimilar medium to be t//«-
formy and to tend perpendicularly to the plane ofthehori"
zon ; to define the motion of a projeStile therein, which f of-
fers refijlance proportional to its velocity. (PI. 1, Fig. 4.)
Let the proje6lile go from any place D in the dire6lion of
any right line DP, and let its velocity at the heginning 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, fo that DA may be to AC as therefiflance
of the medium arifing from the motiop upwards &t the begia-
jDing to the force of gravity; or (which comes to the fame) fo
that the reftangle under DA and DP may be to that under
AC and CP as the whole )reiiftance at the beginning of the
toertion to the force of gravity. With the afymptotes DC,
CP defcribe any hyperbola GTBS cutting the perpendiculars
J)(3, AB in G and B ; complete the pivrallelogram DGKC,
and let its fide GK cut AB in Q. Take a line N in the fame
ratio to QB as DC is in to CP ; and from any point R of the
right line DC erecl RT perpendicular to it, meeting the hy-
perbola in T, and the right lines EH, GK, DP in I, t, and
tCT
y. ; in Jhat perpendicular take Vr equal to -tj-j Qfj, which is
CTTF
the fame thing, take Rr equal to ' -j^. ; and the proje6lile
in the time DRTG will arrive at the point r, defcribing the
£arve line DiaF, the locus of the point r ; thence it will come
to its greatefl height a in the perpendicular AJJ ; and after-
B 3
6 MATHEMATICAL PRINCIPLES Book IL
... . , . . . « • _
wards ever approach to the afymptote PC. And its velocity
in any point r will be as the tangent rL to the curve. Q.E.I.
For N is to QB as DC to CP or DR to RV, and therefore
RV is equal to ^^ ?! ^^, and Rr (that is, RV — Vr, or
. N
DR X QB — tGT> . - DR X AB — RDGT ^r
^ ■ J IS equal to ,^ — — ^ Now
kt the time be expounded by the area RDGT and (by laws, cor.
%) diftinguifh the motion of the body into two others, one of
afcent, the other lateral. And fince the reiiftance is as the
motion, let that' alfo be diftinguiih^ into two parts propor*
tional and contrary to the parts of the motion : and thereibre
the length defcribed by the lateral motion will be (by prop. 2,
book 2) as the line DR, and the height (by prop. 3, booi; 2)
as the area DR X AB — RDGT, that is, as the line Rr.
But in the very beginning of the motion the area RDQT is
equal to the re<9;angle DR X AQ, and therefore that line Rr
. DR X AB — DR X AQ\ .„ ., u * m> at>
(or ^T ) will then be to DR as AB —
, N ^
AQ or QB to N, that is, as CP to DC; and therefore as the
motion upwards to the motion lengthwife at the beginnipg.
Since, therefore, Rr is always as the height, and DR always qm
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 tfarat the body will move in the
line DraF, which is the locus of the point r. Q.E.D.
n . rru c T> 1* DRxAB RDGT
Cor. 1. Therefore Rr is equal to ^^ ^-ttf — •
and therefore if RT be produced to X fo that RX may be
DR X AB
equal to ■ , that is, if the parallelogram ACPY be
completed, and DY cutting CP in Z be drawn, an4 RT be
][>roduced till it meets DY in X ; Xr will be equal to , and
therefore proportional to the time.
Cob. 2« Whence if innumerable lines CR, or, which is
the iame^ ianun),€xable line^ Z}^, be taken in a geometrical
SeS^ I. OP NATIFItAL PHIIXHRXPBY. 7
progr^ffion^ there will be a$ maay lines Xr io an arithnxetidEil
progce^n. And tbence die cunre IkaF is «afily delineated
by the table of logarithms.
Cob. 3. If a parabola be oonftnidted to the vertex D (PI.
i. Fig. 5)y and die diameter D6 produced downwards^ and
its latus re6lum is to ^ DP as the whole refiftance at the be^
ginning of the motion to the gravitating force^ the velocity
with which the body ought to go from theplace D^ in the direo*
tko of the right line DP^ fo as in an unif(»rm refifting medium
to defcribe the curve DraF^ will be the fame as that with which
it ought to go firom the fame place D, in the direction of the
fame right line DP, fo as to defcribe a parabola in a non-refift*
ing medium. For the latus re&um of this parabola, at the
DV* tGT
very beginning of the motion, is -^ — ; and Vr is-rj^ or
DR X Tt
■ • But a right line^ which, if drawn, would touch
the hyperbola GTS in G, is parallel to DK, and therefore Tt .
. CK X DR , T^T • QB X DC . ^ *u r \r •
IS -— , and N is -~ . And therefore Vr is
equal to — o DC* x QB ^ *^* ^^ (becaufe DR afjd DC^ .
DV* X CK X CP
DV and DP are proportioniJii), ^ — 2f)P^ x QB — ' ^^^
DV* iDV^ X QB
the latus re^lum -r^ comes out -77^ — ttd*^ ^^* ^^ (becaufe
Vr * \jfL X v-'l
QB and CK, DA and AC are proportional), -j^ — pp-,and
therefore is to eDP as DP x DA to CP x AC ; that is, as
the refiftance to the gravity. Q.E.D.
Co». 4. Hence if a body be projeAed from any place D
(Pi. 1, Pig. 4) with a given velocity, in the dire<9:ion of a
right Kae DP given by pofition, and the refiftance of the
medium, at the beginning of the motion, be given, ,the curve
DraF, which that body will defcribe, may be found. For the
velocity being given, the latus ie6hmi of the parabola is given^
aa is w«li known. And taking SDP to that latiis reS.iim, as
the ferc^ of gravity to tbe refifting force, DP is alfo given,
6 4
8 MATHSMATICAL PRfDr011>tBB BooklX.
Then cutting DC in A, fo thai CP x AC may be to DP x DA
in the fame ratio of the gravity to the refiftance^ the point
A will be given. And hence the curve DraF is alfo given.
CpB. 5. And, on the contrary, if the curve DraF be givfen,
there will be given both the velocity of the body and the
refiftance 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 refiftance of the medium at the beginning of the
motion, and the latus re6);um of the parabola ; and thence
the velocity at the beginning of the motion is given alfo.
Then from the length of the tangent rL there is given
both the velocity proportional to it, and the refiftance proper**
tionai to the velocity in any place r.
CoR. 6. But fince the length 2DP is to the latus re6lum
of the parabola as the gravity to the refiftance in D ; and,
from the velocity augmented, the refiftance is augmented
in the fame ratio, but the latus re6lum of the parabola, is aug-
mented in the duplicate of that ratio, it is plain that the
length 2DP is augmented in that fimple ratio only ; and is
therefore c^lways proportional to the velocity ; nor will it be
augmented or diminiftied by the change of the angle CDP,
nnlefs the velocity be alfo changed.
Cob. 7. Hence appears the method of determining the
curve DraF (PI. 1, Fig. 6) nej^, from the phagnomena, and
thence coUedling the refiftance and velocity with which the
body is projefted. Let two fimilar and equal bodies be pro-
je6ted with the fame velocity, from the place D, in different
angles CDP, CDp; and let the places F, f, where they fall
upon the horii^ontal plane DC,.be known. Then taking any*
length for DP or Dp, fuppofe the refifl:ance in D to be to the
gravity in any ratio whatfoever, and let that ratio be expound-
ed by any length SM. Then, by computation, from that af-
fumed length DP, find the lengths DF, Df ; and from the ratio
Ff
=r=5 found by calculation, fubdudl the fame ratio as found
DF -^
by experiment; and let the difference be expounded by the
perpendicular MN. Repeat the fame a fecond and a third
time, by affuming always a new ratio SM of the refiftance to
the gravity, and coUefting a new difference MN. Draw the
Se6i. I. OP KATURAL PHILOSOPHY. 9
affirmative diffi^rences on one fide of the right line SM/and
the negative on the other fide; 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 refiftance to the gra-
vity, which was to be found. From this ratio the length DF
is to be collefted by calculation ; and a length, which is to
the aflumed length DP as the length DF known by experi-
ment to the length DF juft now found, will be the true length
DP. This being known, you will have both the curve line
DraF which the body defcribes, and alfo the velocity and re-
fiftance of the body in each place.
SCHOLIUM.
But, yet, that the refiftance of bodies is in the ratio of the
velocity, is more a mathematical hypothefis than a ph5'fical
one. In mediums' void of all tenacity, the refiftances made
to bodies are in the duplicate ratio, of the velocities. For by
the a^lion of a fwifter body, a greater motion in proportion
to a greater velocity is communicated to the fame quantity
of the medium in a lefs time ; and in an equal time, by rea«-
fon of a greater quantity of the difturbed medium, a motion
is communicated in the duplicate ratio greater; and the re-
fiftance (by law 2 and 3) is as the motion communicated.
Let us, therefore, fee what motions arife from this law of re
fiftance. .
SECTION II.
Of the motion of bodies that are refifted in the duplicate ratio
of their velocities,
PROPOSITION V. \ THEOREM III.
If a body is refifted in the duplicate ratio of its, velocity, and
moves by its innate force only through a fimilar ^medium;
and the times be taken in a geon/ietrical progrejjion, proceed"
ing from lefs to greater terms: I fay, that the velocities at
the beginning of each of the time's are in the fame geometrical
progrejfion inverfely; and that the fpaces are equal, which
are defcribedin each of the times. (PI. 1, Fig. 7.)
For fince the refiftance of the medium is proportional to the
iqutu'e of the^ velocity, and the decrement of the velocity is
|MoportionaI to the refiftance : if the time be divided into in-
iO MATHEMATICAL PRINCIPLBS BqoIc IL
fiumerable equal particles^ the fquves of the velocities ot the
b(£gin9ing of each of the times will be proportional to the
^iffisrences of the fame velocities. Let thofe particles of time
})e AKj KL, LM, &c. taken in the right line CD; and ereA
the perpendiculars AB^ Kk, U, Mm, &c, meeting the hyper-
bola BklmG, defcribed with the centre C^ and the re&angu-
)ar afymptotes CD^ CH^ in B, k, \, m, 8cc. ; then AB will be to
Kk as CK to CA, aqd, by divifion, AB — Kk to Kk as AK
io CAf and, alternately^ A3 — Kk to AK as Kk to CA ; and
therefore as AB x Kk to AB x CA. Therefore fince AK
and AB x CA are given, AB -7 Kk will be as AB x Kk;
and, laflly, when AB and Kk coincide, as AB*. And, by the
like reafoning, Kk — U, LI -^ Mm, 8cc. will be as Kk% Ll%
Sac. Therefore the fquares of the lines AB, Kk, LI, Mm, Sec.
are as their differences; and, therefore, fince the fquares of
the velocities were (hewn above to be as their differences, the
pr(^effion of both will be alike. This being demonfirated
it follows alfo that the areas defcribed by thefe Unes are in
a like progreifion with the fpaces defcribed by thefe
velocities! Therefore if the velocity at the beginning of
jthe firft time AK be expounded by the line AB, and the
velocity at the beginning of the fecoii^d time KL by the
line Kk, and the lengtj^ defcribed in the firft time by the
area AKkB, all the following velocities will be expounded by
the following lines LI, Mm^ &c. and the lengths defcribed,
by tbe areas Kl, Lm, &c. And, by compc^tion, if the whole
time be expounded by AM, the fupi of its parts, the whole
length defcribed will be expounded by AMmB the fum of its
parts. Now conceive the time AM to be divided into the
parts AK, KL, LM, &c. fo that CA, CK, CL, CM, 8cc. may
be in a geometrical progreffion; and thofe parts will be in the
fame progreffion, and the velocities AB, Kk, LI, Mm, &c.
will be in the fame progreffion inverfely, and the fpaces ao-
fcribed Ak, Kl, Lm, &c. will be equal. Q.E.D.
Cob. 1. Hence it appears, that if the time be expounded
by any part AD of the afymptote, and the velocity in the
beginning of the time by the ordinate AB, the velocity ajt
the end of the time will be expounded by the ordinate DG;
and the whole fpace defcribed by the adjacent hyperbolic
Se&. II. OF NATURAL PHILOSOPHY. >/ 11
area ABGD; and the fpace which any body can defcribe ia
the fame time AD, with the firft velocity AB, in a non-refift-
ing medium, by the redlangle AB X AD.
Cob. 2. Hence the fpace defcribed in a reiifting medium
is given, by taking it to the fpace defcribed with the uniform
velocity AB in a non-reiifiing medium, as the hyperbolic area
ABGD to the recftangle AB X AD.
Cob. 3. The refiftance of the medium is alfo given, by
ina^cing it ^qual, in the very beginning of the motion, to aa
uniform centripetal force, which could generate, in a body
falling througli a non-refifting medium, the velocity AB in
the time AC. For if BT be drawn touching the hyperbola
in B, and meeting the afymptote in T, the right line AT will
be equal to AC, and will expreis the time in which die firft
refiftance, uniformly continued^ may take away the whol^
velocity AB.
Cor. 4. And thence is alfo given the proportion of this re-
fiftance to the force of gr&vity, or any other given centripetal
force.
Cob. 5. And,viceverfa,i£iheTe is given the.proportion of the
refiftance to any giyen centripetal force, tiie time AC is alfo
given, in which a centripetal force equal to the refiftance may
generate any velocity as AB; and thence is given the point B»
through which the hyperbola, having CH, CD for its afymp-
totes, is to be defcribed; as alfo the ^ace ABGD, which a
body^ by beginning its motion with that velocity AB, can de-
fcribe in any time AD, in a fimilar refifting medium.
PROPOSITION VI. THEOREM IV.
Homogeneous and equal fpherical bodies, oppofed by refiftances
that are in the duplicate ratio of the, velocities, and moving on.
by their innate force only, will, in times which are reciprocal'^
fy as the velocities at the beginning, defcribe equal fpaces,
and lofe parts of their velocities proportional to the wholes.
(PI. 1, Fig. 8.)
To Ae rediangular afymptotes CD, CH defcribe any hy-
perbola BbEe, cutting the perpendiculars AB, ah, DE, de
in B^ b, E, e; let the initial velocities be expounded by the
perpendiculars AB, DE, and the times by the lines Aa, Dd.
It MATHEMATICAL t»ftINCIPLES Book it
Therefore as Aa is to Dd, fo (by the hypothefis) is DE to AB,
and fo (from the nature of the hyperbola) is CA to CD; and,
by coinpofition, fo is Ca to Cd. Therefore the areas ABba,
DEed, that is, the fpaces defcribed, are equal among them-
felves, and the firft velocities AB, DE are proj>ortional to the
laft ab, de; and therefore, by divifion, proportional to the
parts of the velocities loft, AB — ab, DE — de. Q.E.D.
PROPOSITION VII. THEOREM V.
Jf Spherical bodies are rejijted in the duplicate ratio of their
velocities f in times which are as thejirjl motions direSly, and
thejirji refiftances inverfe/j/, they will lofe parts of their mo^
tions proportional to the wholes, and will defcribejpaces pro^
portional to thofe times and the firft velocities conjunSlly.
For the parts of the motions loft are as the reftftances and
times conjun6llj% Therefore, that thofe parts maybe pro^
portional to the wholes, the reiiftance and time conjun^lly
ought to be as the motion. Therefore the time will be as tli^
motion diretftly and the refiftance inverfely. Wherefore th^
particles of the times being taken in that ratio, the bodies
will always lofe parts of their motions proportional to the
wholes, and therefore will retain velocities always propor-
tional to their firft velocities. And becaufe of the given ra-
tio of the velocities, they will always defcribe fpaces which
are as the firft velocities and the times conjun<Sly. Q.E.D. .
CoR. 1. Therefore if bodies equally fwift are refifted in a
duplicate ratio of their diameters, homogeneous globes mov-
ing with any velocities whatfoever, by defcribing fpaces pro-
.portional to their diameters, will lofe parts of their motions
proportional to the wholes. For the motion of each globe
will be as jts velocity and maft conjuncSlly, that is, as the ve-
locity and the cube of its diameter; the refiftance (by fuppofi-
tion) will be as the fqnare of the diameter and the fquare of
the velocity conjun6lly; and the time (by this propoiition) is
in the former ratio dire<Slly, and in the latter inverfely, that is,
as the diameter direftly and the velocity inverfely; and there-
fore the fpace, which is proportional to the tiipe and velocity^
is as the diameter.
J^/a^I.Wll.
<i»
/za^^/£
H
i^/<7/.
R
^^^J
CB A ^ ^ 5
i^ ^ ^ i7 c
irr
^ff^
H
B
AKLMN
H
S^-
H/
AKLMT D
H
jT^a.
C A^ D^
wafrwifMM s«.J/waty4$u Si{J3^)u^*
SeB. IL OF NATUBAI^ PUILOSOPUY. 13
Cor. 2. If bodies equally fwift are refifted in afefqulplicate
ratio of their diameters^ homogeneous globes^ moving with
any velocities whatfoeveo by defcribing fpace» that are in a
fefquiplicate ratio of the diameters^ will lofe parts of their
motions proportional to the wholes.
Cor. 3. And univerfally, if equally fwift bodies are refifted
in the ratio of any power of the diameters^ the fpaces^ in
which homogeneous globes moving with any velocity whatfo-
ever, will lofe parts of their motions proportional to the wholes,
will be as the cubes of the diameters applied to that power.
Let thole diameters be D and E; and if the refiftances, where
the velocities are fuppofed equal> are as D" and E"; the
fpaces in which the globes, moving with any velocities what-
foever, will lofe parts of their motions proportional to the
wholes will be as D^ — » and E^ — «. And. therefore homoge-
neous globes, in defcribing fpaces proportional to I)^-— « and
E^ — ", will retain their velocities in the fame ratio to one ano-
ther as at the beginning.
CoR. 4. Now if the globes are not homogeneous, the fpace
defcribed by the denfer globe muft be augmented in. the ratio
of the denfity. For the motion, with an equal velocity, is
greater in the ratio of the denfity, and the time (by this prop.)
is augmented in the ratio of motion direftly, and the fpace de-
fcribed in the ratio of the time.
CoR. 5. And if the globes move in different mediums, the
fpace, in a medium which, ceteris paribus, refifts the moft,
muft be diminifhed in the ratio of the greater refiftance. For
the time (by this prop.) will be diminiOied in the ratio of the
augmented refiftance, and the fpace in the ratio of the time.
LEMMA II.
The moment of any genitum is equal to the moments of each
of the generating fides drazvn into the indices of tlie powers
of thofe fides, and into their co-efficients continually.
I call any quantity a genitum which is not made^ by addi-*
lion or fubduftion of divers parts, but is generated or pro-,
duced in arithmetic by the multiplication, divifion, or extrac-
tion of the root of any terms whatfoever; in geometry by the
14 - MATHEMATICAL PRINCIPLES Book it.
invention of contents and fides, or of the extremes and means
of proportionals. Quantities of this kind afeprodu6bj quotients,
foots, re6langles, fqtiares, cubes, fqnare and cubic fides, and
th^ like. Tiiefe quantities I here confider as variable and
indetermined, and increafing or decreafing, as it were, by a'
I^rpetual ^motion or finx; and I underftand their momenta*
ileous increments or de<^rements by the name of moments;
fo that the increments may be efieemed as added or affirma-
tive moments; and the decrements as fubdu6ted or negative
ones. But take care not to look upon finite particles as foch.
Finite particles are not moments, bat the very quantities ge-
nerated by the moments. We are to conceive them as the
' jnR. nafcent principles of finite magnitudes. Nor do we in
this lemma regard the magnitude of the moments, but their
firfi; proportion, as nafcent. It will be the fame thing, if, in-
fiead of moments, we ufe either the velocities of the incre-
ments and decrements (which may alfo be called the motions,
mutations, and fluxions of quantities), or any finite quantities
proportional to thofe velocities. The co-efficient of any ge-
- nerating fide is the quantity which arifes by applying the geni-
tum to that fide.
Wherefore the fenfe of the lemma is, that if the moments
oT any quantities A, B, C, &c. increafing or decreafing by a
perpetual flux, or the velocities of the mutations which are
proportional to them, be called a, b, c, 8cc. the moment or
mutation of the generated reftangle AB will be aB + bA;
the moment of the generated content ABC will be aBC'+
bAC + cAB;* and the moments of the generated powers A*,
A% A^ A^ A^, A^, A^, A-", A-% A ~ ^ will be 2aA, 3aA*,
4aA^ f aA — ^, faA^, faA - ^, faA - ^, — aA -^ *, — 2aA-^,
— |aA— ^refpe6lively; and, in general, that the moment
of any power A-, will be - aA "^ , AHb, that the mo-
tfient of the generated quantity A*B -will be 2aAB + bA* ;
the moment of the generated quantity A'B*C* will . be
SaA^B^C* + 4bA3ffG* + 2cA'B*C; and the moment of the
«
A'
^nemted quantity gj or A^B — * will be SaA*B-** —
^fbA'B — ^ ; and fo on. Hie lemma is thus demonftratecl'.
Casb 1. Any redlaogle^ as AB^ augmented by a perpetual
fluXj when^ as yet^ there wanted of the fides A and B half
their moments ia and ^h, was A — |a into B — |b> or ABj
— f a B — |b A + iab; but as foon as the fides A and B
are augmented by the other half moments^ the redlangle be-
comes A + fa into B + |b, oa: AB + |a B + |b A + I ab.
From this re6langle fubdu6l the former redUngle^ and there
will remain the excefs aB + bA. Therefore with the whole
mcremetits a and b of the fides^ the increment aB + bA of
the re£kang\e is generated. Q.E.D.
Case 2. Suppofe AB always equal to G^ and then the
moment of the content ABC or GC (by cafe 1) will be gC +
cG^ that is (putting AB and aB + bA for G and g)^ aBC +
bAC + cAti, And the reafoning is the fame for contents
under ever fo many fides. Q.E.D.
Case 3. Suppofe the fides A^ B^ and C, to be always equal
among themfdves; and the moment aB+bA^ of A% that is^ of
the rediangle AB, will be 2aA ; and the moment aBC + bAC
+ cAB of A^ that is, of the content ABC, will be 3aA*. And
by the fame reafoning the moment of any power A° is
naA»-*. Q.E.D.
Gasb 4. Therefore -fince -r into A is 1, the moment of
A '
, ^ drawn into A, together with -j drawn into a, will be the
moment of 1, that is, nothing. Therefore the moment of
1 . —a 1
■jj or of A— ', is -^» And generally, fince -t^ into A* is 1^
the moment <^-rt; drawn into A* together with -r^iiito naA'
wHl be nothitrg. And, therefore, the moment of -r- or A-^»
wHf be— ^2-» Q.E.D.
16 MATHEMATICAL PRINCIPLES B(H>k IL
Case 6. And fince A into A^ is A, the moment of A^
drawn into 2 A^ will be a (by cafe S); and , therefore, the
uboment of A^ will be -r^ ^^ |aA— *. And^generally, put-
t an
ting A n equal to B, then A« will be equal to B% and there-
feAre maA« — ' equal to nbB» — *, and maA — * equal to
m mm — n
nbB — % or nbA*— n ; and therefore T aA ^is equal to b.
m
that is, equal to the moment of AIT. Q.E.D.
Case 6. Therefore the moment of any generated quantity
A^B** is the moment of A". drawn into B", together with the
moment of B" drawn into A"*, that is, maA«— ' B" + nbB"— '
A" ; and that whether the indices m and n of the powers be whole
numbers or fradlions, afHrmative or negative. And the rea«*
foning is the fame for contents under more powers. Q.E.D.
Cor. J. Hence in quantities continually proportional, if
one term is given, the moments of the reft of the terms will be
as the fame terms multiplied by the number of intervals be-
tween them and the given term. Let A, B, C, D, E, F, be
continually proportional; then if the term C is given, the
moments of the reft of the terms will be among themfelves
as — 2A^ — B, D, 2E, 3F.
CoR. 2. And if in four proportionals the two means are
given, the moments of the extremes will be as thofe extremes.
The fame is to be underftood of the fides of any given reflan-
gle. .
CoR. 3. And if the fum or difference of two fquares is
given, the moments of the fides will be reciprocally as the
fides. ^
SCHOLIUM.
In a letter of mine to Mr. J. Collins, dated December 10,
1672, having defcribed a method of tangents, which I fuf-
pedled to be the fame with S/m/Jms's method, which at th»t
time was not made public, I fubjoined thefe words: This is
one particular, or rather a corollary, of a general method,
which extends itfelf, mthout any troublefome calculation, not
iSfeiS. IF. OF NATtJRAL PHILOSOPHY^ 1?
oHfy to iHedrtming of tangents to any curve lines, whether
geomttricfil or mechanical, or any how refpeSiing right lines or
other curves, but alfo to the refolving other ahftrufer kinds of
problems ab6ut the crookednefs, areas, lengths, centres of gravity
if curves 8cc.; nor is it (as Hudden's method de Maximis
& Minimis) limited to equations which are free from furd
quantities. This methad I have interwoven with that other of
working in equations, by reducing them to infinite f cries. So
for that letter. And thefe laft words relate to a treatife I com-
pofed on that fubjeft in the year 1671. The foundation of
that general method is contained in the preceding lemma.
PROPOSITION VIIL THEOREM VI.
If a body in an uniform medium, being uniformly d^ed upon
by the force of gravity, afcends or defcends in a right line ;
' and the whole fpace defcribed be dijiinguijhed into eqUat
' parts, and in the beginning of each of the parts (by adding or
fubdu6iing the rejijling force of the medium to or fromifle
force of gravity, when the body afcends or defcends) you coU
hit the abfolute forces ; I fay, that thofeahfofute forces arts
in a geometrical progrejfion, (PL 2^ Rg. 1.) '
For let the force of gravity be expounded by the given line
AC ; tbe 'f6rce of refiftance by the indefinite line AK ; the
abfolute^ force in the defcent of the body by the difference
KG ; the velocity of the body by a line AP, which fhall be^^*
mean proportiobal betwe^n^ AK and AGp and ttiirefore ?n? a
fubduplicteitfe Vatio of the i»^iffanc6;= the tt^emfetf t of th^r^
fiftance made in a given ' particle of iitof^hy Ihte lineolae Kfi)
and tb^ contemporaneous increment of 'the Velocity' by the*
Ijneolse PQ ; and with the centre C, and i^cSarigular afymp^
totes CA, CH, defcribe ihy hyperbola BNS meeting the
erefted perpendSculkrs AB>'KN, LO in B, N, and O. Be-
caufe AK is as AP% the moment KL of the one will be as the
moment £APQ of the other, that is,as AP x EC ; for the in-
crement PQ of the velocity is (by law 2.) proportional to the
generating force KC. L^t the ratio of KLbe compounded
with the ratio of KN, and the recftangle Kt x KN will be-
come as AP X KC X KN; that is (becaufe the re6l-
«ngle KC X KN is giveri), as AP. But the ultimate
Vol. IE ' C
18 MATHEMATICAL PRINCIPLES Book II.
ratio of the hyperbolic area KNOLto the rectangle KL
X KN becomes^ when the points K and L coincide, the
rafio of equality. Therefore that hyperbolic evanefcent area
is as AP. Therefore the whole hyperbolic area ABOL is
compofed of particles KNOL which are always proportional
to the velocity AP ; and therefore is itfelf proportional to the
fpace defcribed with that velocity. Let that area be now
divided into equal parts^ as ABMI^ IMNK^ KNOL^ &c. and
the abfolute forces. AC^ IC^ KCj UC, &c. will be in a geo-
metrical progreffion. Q.E.D. And by a like reafon^ng^
in the afcent of the body^ taking, on the contrary iide of the
point A, the equal areas ABmi^ imnk, knol, &c. it will ap-
pear that the abfolute forces AC, iC, kC, IC, &c. are conti-
nually proportional. Therefore if all the fpaces in the afcent
and defcent are taken equal, all the abfolute forces IC, kC,
iC, AC, IC, KC, LC, &c. will be continually proportional.
Q.E.1>.
. Co^. J . H^ce if the fpace defcribed be expounded by
the hyperbolic area ABNK, the force of gravity, the velocity
of the body, and the refiflance of the medium,may be expound-
ed by the lines AC, AP, and AK refpediively ; and vice verfa,
GoR. 2. And the greateft velocity which the body can
eyer acquire in an infinite defcent will be expounded by the
Ij^ AC.
; CpR. S.TberjBfore if the refiftance of the medium anfwering
to any given vel<>Gity be known, the greateft velocity will be
foUDd, by takiqg it to that given velocity in a ratio fubdupU-
cate of the ratio vfhich the force of gravity bears to that
knp^n refiftance of the medium.
PROPOSITION IX. THEOREM VII.;
Suppojing what is above demonftratedylfay, that if the tangents
of the angles ofthefe&or of a circle, and of an hjfperbala,
be taken proportional to the velocities, the radius being of a
fit magnitude, all the time of the afcent to the higheft place
will be as thefeSior qfthe circh, and all the time ofdefcend-
ir^ from the higheji place as the fcBor of thi^ hyperbola.
(Pl.2, Fig..a.)
To tb^e right line AC, which exprefles the foirce of gravity^
let AD be drawn perpendicular Und equal. From the centre
D with the femi-diameter AD defcribe as well the <]^uadrant
S^Si. II. OF NATI3EAL PHILOSOPHY. ^9
AtE of a circle, as the re6langular hyperbola AVZ, whofe
a^^is is AK, principal vertex A, and afymptote DC. Let Dp,
DP be drawn; and the circular feftor AtD will be as all the
time of the afcent to the higheft place ; and the hyperbolic
feiSlor ATD as all the time of defcent from the higheft place ;
if fo be that the tangents Ap, AP of thofe fedors be as the
velocities. (PL 2, Fig. Q,)
Case 1. Draw Dvq cutting oflF the moments or leaft par-
ticles tDv and qDp, defcribed in the fame time, of the feftor
ADt and of the triangle ADp. Since thofe particles (becaufe
of the common angle D) are in a duplicate ratio of the fides,
tJie particle tDv will be as -^-^fu > that is (becaufe tD
is given), as ^i But pD^ is AD» + Ap% that is, AD* +
AD X Ak,o^- AD x Ck; and qDp is | AD x pq. There-
fore tDv, the particle of the fedlor, is as -S- ; that i^, as the
leaft decrement pq of the velooity diredlly, and the force Ck
which diminilhes the velocity, inverfely; and therefore as the
paiticle of time anfwering to the deci'ement of the velocity
And, by compofition, the fum of all the particles tDv in
the fedor ADt will be as the fum of the particles of time
anfwering to each of the loft particles pq of the decreafing
velocity Ap, till that velocity, being diminiihed into nothing,
vapiflies ; that is the whole fedor ADt is as the whole time
of afcent to the higheft place. Q.E.D.
Case 2. Draw DQV cutting off the leaft paiticles TDV
and PDQ of the fedor DAV, and of the triangle DAQ ; and
thefe particles will be to each othel« as DT* to DP*, that is
(if TX and AP are parallel), as DX* to DA* or TX» to AP* \
and, by divifion, as DX* -^ TX* to DA* — AP*. But, from
the nature of the hyperbola, DX* — TX* is AD*; and, by
the fqppofition,AP* isAD X AK. Therefore the particles are
to each other as. AD* to AD^ — AD x AK; that is, as AD
to AD — AK.or. AC to CK : and therefore the particle TDV
of the feiftor" is -' .^^ ; and therefore (becwC? AC
C a
^ HA THEMATIC AL rUlNClPLES JBooklh
PQ
and AD are given) as-—-.; that is^ as the increment of the
velocity diredUy^ and as the force generating the increment
inverleiy ; and therefore as the particle of the time anfwering
to theincrement. And, hy competition^ the fiim of the particles
of time, in which ail the particles PQ of the velocity AP are
, generated, will be as the fum of the particles of the fe^r
ATD ; that is, the whole time will be as the whole fedtor.
Cor. 1. Hence if AB be equal to a fourth part of AC, the
{pace which a body will defcribe by falling in any time will:
be to the fpace which the body could defcribe, by moving
uniformly on in the fame time with its greateft velocity AC,
as the are* ABNK, which exprefles the fpace defcribed in
falling to the area ATD, which exprefTes the time. For fince
AC is to AP as AP to AK, then (by cor. I, lem. 2, of this book)
LK is to PQ as Q.AK to AP, that is, as 2AP to AC, and
thence LK is to iPQ as AP to |AC or AB ; and KN is to AC
or AD asAB txx CK; and, therefore, ex aquo, LKNO to
DPQ a^ AP to CK. But DPQ was to DTV as CK to AC •
Therefore, ex tequo, LKNO is to DTV as AP to AC ; that is,
as the velocity of the falling body to the greateft velocity
whieh the body by falling can acquire. Since, therefore, the
moments LKNO ^d DTV of the areas ABNK and ATD are
as the velocities, all the parts of thofe areas generated in the
fame time will be as the fpaces defcribed in the fame time ;
and therefore the whole areas ABNK and ADT, generated
from the beginning, will be as the whole fpaces defcribed from
the beginning of. Uie defcent. Q»E.D.
Co^. % The lame is true alfo of the fpace defcribed in
the afc^nt* That is to fay, that all that fpace is to the fpace
defcribed in the fame time, with the uniform velocity AC, as
the area ABP,k is to the fe&or ADt.
Co», 3, The velocity of thebody^ falling in the time ATD,
is to the ve}ppity which it would acquire in the fame time iu
^ non-refifting J^ace, as the ti-iangle APD to the hyperbolic
fe<^or ATD. For the velocity- in a non-re0fting medium
would be as the time ATD, mi in » i^eiiftiog medium 19 as
Se£l* IL OP NATURAL PHILOSOPHY. SI
AP^ that is, as the triangle APD. And thofe velocities, at
the beginning of the i)efcent, are equal among themfelves,
as well as thofe areas ATD, APD.
Cor. 4. By the fame argument, the velocity in the afcent
is to the velocity with which the body in the fame time, in a
hon-relifting fpace, would lofe all its motion of afcent, as th^
triangle ApD to the circular fedlor AtD ; or as the right line
Ap to the arc At.
Cor. 5. Therefore the time in which a body, by fulling in
a reiifting medium, would acquire the velocity AP, is to th6
time in which it would acquire its greateft velocity AC, by
falling in a ndn-refiiting fpace, as the fe6lor ADT to the tri-
angle ADC: and the time in which it would lofe its velocity '
Ap, by afcending in a refifting medium, is to the lime in which
it would lofe the fame velocity by afcending in a non-reiifting
fpace, as the arc At to its tangent Ap.
Cor. 6. Hence from the given time there is given the
fpace defcribed in the afcent or defcent. For the greateft
velocity of a body defcending in injimtum is given (by corol.
2 and 3, theor. 6, of this book)'; and thence the time is
given in which a body would acquire that velocity by falling
in a no»-refifting fpace. And taking the feftor. ADT or
ADt to the triangle ADC in the ratio of the given time to
jbhe time juft now found, there will be given both the velocity
AP or Ap, and the area ABNK or ABnk, which is to the
fe6ior ADT, or ADt, as the fpace fought to the fpace which
would, in the given time, be uniformly defcribed with that
greateft velocity found jtrft before.
CoR. 7. And by going backward, from the given fpace of
afcent or defcent ABnk or ABNK, there will be given the time
ADt or ADT.
PROPOSITION X. PROBLEM III.
Suppofe the miiform force of gravity to tend dirt&bf to ike
plane ofth^ horizon, and the refinance to be as the denfity of
the medium andthefqnare 0fihe velocity corgun&ly: itds
propofed to find the dtf^ty of the nvediumiri each place,
which Jhflll make the bpdy^move in anygii;m curse Hue; the
ffelocity of the body and the refiftance of the medium in
each place. (PI. 2, Fig. 3.)
C 3
■i
£d MATHEMATICAL PRINCIPLES Book II.
LetPQ be a plane perpendicular to the plane of the fcheme
itfelf ; PFHQ a curve line meeting that plane in the points
P and Q ; G^ H^ I, K four places of the body going on m
this curve from F to Q; and GBj HC^ ID, K£ four parallel
ordinates let fall from thefe points to the horizon^ and {land-
ing on the horizontal line PQ at the points B, Cj D^ £ ; and
let the diftances BC, CD, DE, of the ordinates be equal
among themfelves. From the points G and H let the right
lines GL, HN, be drawn touching the curve ih G and H,
tod meeting the ordinates CH, DI, produced upwards, in L
and N; and complete the parallelogram HCDM. And the
times in which the body defcribes the arcs GH, HI, will
be in a fubduplicate ratio of the altitudes LH, NI, which the
bodies would defcribe in thofe times, by falling from the tan-
gents; and the velocities will be as the lengths defcribed GH,
HI duredily, and the times inverfely. Let the times be ex-
CH HI
pounded by T and t, and the velocities by — =- and — ; and
A t
the decrement of the velocity produced in the time t will be
CH HT
expounded by -7= . This decrement arifes from
It
the refiftance which retards the body, and from the gravity
which accelerates it. Gravity, in a faUing body, which in
its fall defcribes the fpacc NI, produces a velocity with which
it would be able to defcribe twice that fpace in the fame time,
2NI
as Galileo has demonftrated ; that is, the velocity — — : but if
I t
the body defcribes the arc HI, it augments that arc only by
the length HI — HN or . ^ ; and therefore generates
xJi
2MI X NI
only the velocity — |Tf—- Let this velocity be added
to the before-mentioned decrement, and we Ihall have the
decrement of the velocity ariflng from the refiftance alone,
. . GH HI ^ 2MI X NI ^, . ^
that IS, -7= -^ — H — trr^- Therefore fince, m
1 tr t X Jtn
•the fame time, the adion of gravity generates, in a falling
SeS. n. OP NATURAL PHILOSOPR-7. 9A
body, the velocity , the refiftance will be to the gravity
GH HI , SMI X NI 2NI t X GH „,
as — ■■ — .ii— -I- —.• to . i Of as — xli
T t ^ t X HI ^^ T
+ pr to 2NI.
Now for the abfciffas CB, CD, CE, put — o, o, 2o. For
the ordinate CH put P ; and for MI put any feries Qo + Ro»
+ So', &c. And all the terms of the feries after the firft, that
is, Ro* + So' +, 8cc. will be NI ; and the ordinates DI, EK,
and BG will be P — Qo — Ro* — So' — , &c. P — 2Q0 —
4R0* — 8S0' — , &c. and P + Qo — Ro* + So' — , &c. re-
fpediively. And by fquaring the differences of the ordinates
BG — CH and CH — PI, and to the fquares thence pro-
duced adding the fquares of BC and CD themfelves, you will
have 00 + QQoo — 2QR0' +, &c. and 00 + QQoo +
2QR0' +, &c. the fquares of the arcs GH, HI ; whofe roots
, QRoo QRoo
are the arcs GH and HI. Moreover, if from the ordinate CH
there be fubduded half the fum of the ordinates BG and DI,
and from the ordinate DI there be fubdudted half the fum of the
ordinates CH and EK, there will remsdn Roo and Uoo +
3S0', the verfed fines of the arcs GI and HK. And thefe are
proportional to the lineote LH and NI, and therefore in
the duplicate ratio of the infinitely fmall times T and t:
^1 * .u .' t . ^ R + iiSo R + |So ^ r
and thence the ratio 757 is • -r or 5 ; »»«
T, R R
tjc^GH _ jj J ^ QMI X NI^ ^y fubftituting the values of
—, GH, HI, MI and NI juft found, becomes ^^v^i+QQ.
T 2R
And fince 2NI is 2R00, the refiftance will be nqw to the gravi-
3S00
ty ^ -gR" Vl + QQ to 2R00, that is, a»sS v^l + QQ to
4RR. C 4
ji .
34 MATHEMATICAL PRINCIPLES BooklU
And the velocity will be fuel), that a body going off there-
with from any place H, in the dire6lion of the tangent HN,
H'ould defcrihc, in vacuo, a parabola, whofe diameter is HC,
, . , ■ '• HN* r+QQ
and Its latas rectum -rrr- or ^ ,. •
NI R
And the refiftance is as the denfity of the medium and the
fijoareof the velocity conjunAly; and therefore the denfity of
the medium is as the refiftance dire6ily, and the fquare of
the velocity inverfely; that is, as ^L^iL+S^direaiy and
1 -L OO - S
\^^ inverfely ; that is, as Q.E.I.
R R V 1 4- QQ
Cor. 1. If the taiigent HN be produced both ways, fo as
HT __
to meet any ordinate AF in T,-— < will be equal to • 1 +QQ,
and therefore in what has gone before may be put for • 1 + QQ.
By this means the refiftance will be to the gravity as 3S x HT
HT
to 4RR X AC ; the velocity will be as -j-s, --, and the den.
AC \/ R
S X AC
fity of the medium will be as — =,•
Iv X ili
CoE. 2. And hence, if the curve line PFHQ be defined
by the relation between the bafe or abfciffa AG and the or-
dinate CH, as is nfual, and the value of the ordinate be
refolved into a converging feries, the problem will be expe-
ditioufly fcJved by the firft terms of the feries ; as in the fol-
lowing examples;
Example 1. Let the line PFHQ be a femi-circle defcrib-
ed upon the diameter PQ, to find the denfity ^ the medium
that (hall make a projectile move in that line.
Bifeft the diameter PQ in A; and call AQ, n ; AC, a;
CH, e; and CD, o: then DP or AQ» —AD* = nn —
aa — 2ao — oo, or ee — 2ao — oo ; and the root being ex-
tradled by our method, will give DI =: e— ^ J
ao oo
e 2e
aaoo ao' a' o'
"ii^ "gi^ "o^
a o
— — , ?cc. Here put on for ee + aa, and
St&p III OF NATURAL' PHILOSOPHY. ^5.
^^ .„ , ao nnoo anno'
DIwillJ)ecome =e — -.^-^^ -^ — , 8cc.
Such feries I diftinguifh. into fucceffivei terms after tbi{|
manner: I call that the firft term in which the infiniteljr
fmall quantity o is not found; the fecond^ in which that
quantity is of one dimenfion only ; the thirds in which it arties
to two dimenfions ; the fourthj in which it is of three; and (b
ad it^nitum. And the firft term, which here is e, will alwayf
denote the length of the ordinate CH, (landing at the beginT
ning of the indefinite quantity o. Tlie fecond term, which
ao
here is — 5 will denote the diflference between CH and DNt
e . .^
that is, the lineolas M N which is cut off by completing the
paralleldgram HCDM ; and therefore always determines th^
pofition of the tangent HN; as, in this cafe, by taking MN
aO'
to HM as — to o, or a to e. The third term, which here
e
is — ^j will- reprefent the lineolas IN^ which lies between th^
•ge
tangent and the curve ; and therefore determines the angle of
conta6i IHN, or the curvature which the curye line has in H.
If that lineolee IN is of a finite magnitude, it will be exprefre4
by the third term, together with thofe that follow in.it^Uum.
But if that lineolse be diminiflied in infinidum, the terms foUowr
ing become infinitely lefsthan the third term> and therefore
may be negle<5^ed. The fourth term determines the variation
of the curvature ; the fifth, lihe variation of the variation ; and
ib on. Whence, by the waiy, appears no contemptible ufe
of thefe feries in the folution of problems that depend upon
tangents, and the curvature of curves.
^, ' - . ao nnoo anno* ^ . ,
Now compare the fenes e-— — — S^"~""^"l &c.wita
e tSxt use
*the feries P — Qo -- Roo —-So' — &c. and for P, Q, Rand
a nn , ann
c^^ P""*^' e' W^^ i?' ^""^ ^ i+QQput 1 + —
n • ' a
or - J and the denfity of tha medium will come out as — •
46 MATftEMATlOAL PRINCIPLES Book II,
a AC
that is (becaufe n is given), as -J or ^^77' that is, as that length
of the tangent HT, which is terminated at the femi-diameter
AF {landing perpendicularly on PQ : and the refiftance will
be to the gravity as da to 2n, that is, as sAC to the diameter
PQ of the circle ; and the velocity will be as \/ CH. There-
fore if the body goes from the place F, with a due velocity, in
the direction of a line parallel to PQ, and the deniity of the
medium in each of the placed H is as the length of the tan-
gent HT, and the refiftance alfo in any place H is to the force
of gravity as SACto PQ, that body will defcribe the quadrant
FHQ of a circle. Q.E.I.
But if the fame body fhould go from the place P, in the di-
re^lion of a line perpendicular to PQ, and fhould begin to
move in an arc of the femi-circle PFQ, we muft take AC or a
.on the contrary fide of the centre A ; and therefore its figu
muft be changed, and we muft put — a for -[- a. Then the
o
denfity of the medium would come out as • But nature
does not admit of a negative denfity, that is, a denfity which
accelerates the motion of bodies ; and therefore it cannot na-
turally come to pafs that a body by afcending from P fhould
defcribe the quadrant PF of a circle. To produce fuch an
eHeft, a body ought to be accelerated by an impelling ine-
dium, and not impeded by a refifting one.
Example 2. Let the line PFQ be a parabola, having its
axis AF perpendicular to the horizon PQ, to find the denfity
of the medium, which will make a proje6iile mqve in that
line. ' (PI. 2, Fig. 4).
From the nature of the parabola, the redbmgle PDQ is
equal to the re6langle under the ordinate DI and fome given
right line: that is, ifthat right line be called b ; P^C, a; PQ,
c ; CH, e ; and CD, o ; the re6langle a + o into c — a — o
.or ac — aa — 2ao *+ co — 00, is equal to the re6langle b into
'rv» ,' « T^T . 1 ac — aa c — 2a 00
DI, and therefore DI is equal to — r h "^"^ ^ — "T*
. *^ c 2a *
Now the fecond term — r — o of this feries is to be put for
SeSl. II. OF NATITRAL PHItOSOPRY. 27
Qo^ and the third term -r- for Roo. But fince there are no
more terms^ the coefficient S of the fourth term will vanifh;
S
and therefore the quantity ? to which the den-
^ Ry 1 +;QQ
fity of the medium is proportional, will be nothing. There-
fore, where the medium is of no denfity, the projeftile will
move in a parabola; as Galileo hath heretofore demonftrated,
Q.E.L
Example S. Let the line AGK be an hyperbola, having
its afymptote NX perpendicular to the horizontal plane AK,
to find the denfity of the medium that vrill make a projed^ile
move in that line. (PL 2, Fig. 5.)
Let MX be the other afymptote, meeting the ordinate D6
produced in V ; and, from the nature of the hyperbola, the
redlangle of XV into VG will be given. There is alfo given
the ratio of DN to VX, and therefore ^ the redlangk of DN
into VG is given. . Let that be bb : and, completing the pa-
rallelogram DNXZ, let BJif be called a ; BD, o ; NX, c; and
let the given ratio of YZ to ZX or DN be — • Then DN will
bb
m
be equal to a i— o, VG equal to t VZ equal to — x a— o,
and GD or NX — VZ -^VG equal toe— ^ a + '^ o~
^ n n
• Let the term - v ■ be refolved into the converging
^ . bb bb bb ' bb , , ' ^^ .n ^
fenes h — o + ~-oo + -^o3, 8cxj. and GD will be-^
a aa a^ a*
, m bb m bb bb ^
come equal toe a + —o o r*o* —
, . n . a n aa a*
-r o*, &c. The fecopd term — • o — — o of this feriesis,!©
•a* • .. F ; .,; n • - aa ■ ^
Kh
-be ufedfor Qo; the third --r o*, with its fign chang^ f^r
bb
Ro*; and the fourth -^ o\ with its fign phanged alfo for So'^
£8 MATHEMATICAL FSINCIPLB8 JSook IT^
, , . n. . ^ bb bb ,bb
and their coefficients — — — > — ? and - - are to be put for
n aa a a^ *
jQy R, and S in the former rule. Which being done^ the den-
bb
fity of the medium will come out as
a*
bb^/ ™°^ 2mbb b*
aJ" nn naa a*
or — . , J ? that is, if in VZ you take
^ mm 2mbb b*
^ aa + aa 1
nn n aa
',r^r 1 . ^Tr^ 1 -r.. ,01*. 2mbb M
VY equal to y G, as tt^* F6r aa and -r a ^ —
^ XY n* n aa
are the fquares of XZ and ZY. But the ratio of the refift-
ance to gravity is found to be that of SXY to 2YG ; and the
velocity is that with which the body would defcribe a para-
XY*
bola, whofe vertex is G, diameter DG, latus re6lum*-T™-
Suppofe, therefore, that the denfities of the medium in each
of the places G are reciprocally as the diftances XY, and that
the refiftance in any place G is to the gravity as dXY to
gYG ; and a body let go from the place A, with a due velo-
city, will defcribe that hyperbola AGK. Q.E.I.
Example 4. Suppofe, indefinitely, the line AGK to be an
hyperbola defcribed with the centre X, and the afymptotes
MX, NX, fo that, having conftruAed the re6langle XZDN,
whofe iide ZD cuts the t^yperbola in G and its afymptote in
V, VG may be reciprocally as any power DN"* of the line ZX
or DN, whofe index is the number n : to find the denfity of
the medium in which a proje6led body will defcribe this
curve. (PL 2, Fig. 5.)
For BN, BD, NX, put A, O, C refpedively, and let VZ be
to XZ or DN as d to e, arid VG be equal to —i,; then DN
bb
%ill be equil to A — O, VG =: ■-' \ 9 VZ = - A— O,
^ A — 0|' ^ e
4ridt3fDorNX — VZ— VGequaltofc — ^A +- O —
e 6
£=:♦ Let the term ===- berefdved into an mfinit^
A -or A— op
bb . nbb nn + n , , ^. , n^ + 3nn+2n
X bb 0*, Sec. and GD will be equal to C A — ^+lo
^ e A" e -
nbb ^ + nn + n , v ^. " + n» + 3nn + an , , _.4
-afp^ — sa»t^ ^^ ^ 6A^T^ — :^^^-.
d nbb "
&c. The fecoiwi term - O — -ttti O of this feries is to be
e A"+
ufed for Qo, Ihe third ^^ ,^ bbO* for Roo, the fourtk
• TTTrjTr — bbO' for So\ And thence tlie den$)y>
S'. . ' ..: f
of tb^ u^uedium itJ^V ^.'Ka^ ^^ ^^ P^^^ ®^ will, bj^
I f • .»
— . ,. .,, .i > J'.u I u A^. and t)hcrefipre if ia V2i
yoti ta^e VY* equal to n X V(5r, that denfity is reciprocally
vx^ X. *. ,dd'. €dnbb . nnb* , " . ;
9^ -?iY* . For A* and -r- A* -^ Va- " -^+"T3r^'*^ thefijuares^
of XZ and ZY. But the refiftance in the fame place G is tq
XY
ibc force of gravity as SS x -r- to4RI{7 that is, as^ XY tq
2ntt + 2n
" ' ' ■'-■^-' t VG. And the velocity there is the fame wheret
n + 2 . ^
with tb^. projected body would move i;i a pcirabola, whole ver^
tex is G, diameter GD, andlatus reftum — ^r- — or
R nn+nxVG*
Q.E.L
SCHOLIUM,
In the fame manner that the denfity of the medium cpn^s
S xAC
out to be as -^ — ttt^j in cor. 1, if the refiftance is put as
li X xli
any power V* of the velocity V, the denfity of the medium
30 . MATBSMATICAt PKINCIPLES Book II.
(Fig. 3.)
•II , S AC ""^ ^
Will come out to be a^^ ^- -^^
R^ HT
Aiid therefore if a curve cante fjaujid, fucli that the ratio of,
n — 1 S* 1
5 or of gTn- to 1 + QQI*""" ' maybe
S
to
HT
R-T- AC
given ; the body^ in an uniform medium^ whofe refiftanoe is
as the power V^ of the velocity V, will move in this curve.
But let us return to more fimple curves.
Becaufe there can be no motion in a parabola except in a
non-relilling medium^ but In the hyperbolas here defcribed
it is produced by a perpetual reiiftance ; it is evident that the
line which a projedlile defcribes in an uniformly refifting me-*
dktin approaches nearer to thefe hyperbolas than to a para-
bola. That line is certainly of the hyperbolic kind^ but about
ihie vertex it is more diftant from the afymptotes^ and in the
parts remote from the vertex draws nearer tP them than thefe
^perbolas here defcribed. The-diflference, however, is not fo
great between the one and theother but that thefe Utt^may
b^; conamodioufly enougji ufed . ip praAice inflead of the for^*
m'ef." And perhaps thefe miiy prbv^ moie ufeful ihan an
hyperbola that is more accurate, and at the fame time more
comppunded. They may. be ms^de ufe of, then, iu^this^n^aa-
ner. (PI. 2, Fig. 5.)
i Gt)mplete the parallelcgram XYGT, and the right line OT
will touch the hyperbola in Q, and therefore the denfity of
the medium^ in G is. reciprocally as tlie tangent GT, and the
GT* • '
velocity there as • -prrr 5 and the refiftance is to the force
r •. /^.rr . 2nn + Sn
of gravity as GT to ~— t-t— x GV.
Therefore if a body projected from the place A, in the direc-
tion of the right line 'AH (PL 2, Fig. 6), defcribes the hyperbola
AGK, and AH produced me^ts theafymptote NX in B;and
Al drawn parallel to it meets the other afymptote MX ijnil.;
the denfity of the medium in A will be reciproca% as AH,
AH* \ L -
and the velocity of the tody as \/ "TT' ^"^ ^^^ refiftance
tScS. 11.' Of. NA3*URAL P9IL0S0PHY. 31
there to the force of gravity as Aji to , - X AI.
Hence the following rules are deduced.
RuLtf 1. If the denfity of the medium at A, and the velo-
city with which the hody is projedled remain the fame, and
the angle NAH be changed ; the lengths AH, AI, HX will
remain. Therefore if thofe lepgths, in any one cafe, are found,
the hyperbola may afterwards be eafily determined from any
given angle NAH. ;
Rule 2. If the angle NAH, and the denfity of the me*
dium at A, remain the fame, and, the velocity with which the
body, is proje&ed be changed, the length AH will continue
the fame ; and AI will be changed in a duplicate ratio of th^
velocity reciprocally. '
Rules. If the angle NAH, the velocity of the body at
A, and the accelcralive gravity remain' the fame, and the pnv
portion of the refiftance at A to the motive gravity be aug^.
ment^ in any ratio ; . the proportion of AH to AI will be aug-^>
men ted in the fame ratio,' the'latus re6lum of the above t
mentioned parabcda remaining the fame, and alfo the length
AH* . •
-jT- proportional to it ; and therefore AH will be dimini(he<i
in the fame rsttio, and AI will be diminiihed in the dupUcate
of that ratio. But the proportion of the refiftance to the
weight is laugmented, when either the fpecific giavily is made
left; the m^gnitudie remaining ^ual, or wheii the denfity of
the ined^ppi is made greater, or when, by diminifliing the
magnitude, the refiftance beconies diminilhed in a lefs ratio
than the 'wdght.
Rule 4. Becaufe the denfity. of the mediuin is greater near
the vertex of the hyperbola than it is in the place A, that a
mean denfity may be preferved, the ratio of the leaft of the
tangents GT to the tangent AH ought to be found, and the
denfity in A augmented in a ratio a little greater than that
of half the fom of thofe tangents to the leaft of the tan-
gents GT.
Rule 5. If the lengths AH, AI are given, and the figure
AGK is to be defcribed, produce HN to X, fo that H^ may
ii MATHEMATICAL niNCI^LBi ' Sook If.
he to AI as Ti' + 1 to 1 : and with the centre X. and the
afymptotes MX, NX>^elcribe an hyperbola through the
point A, fiich that AI may be to any of the Unes VG as XV«
toXK r
Rule 6. By how much the greater the number n is, iCb
much the more accurate are thefe hyperbolas in the afdtnt of
the body frpm A, and lefs accurate in its defeent to K ; and
the contrar}'. The conic hyperbola keeps a meaTi ratio be-
tween thefe, and is more iimple than the reft. Therefore if
the hyperb<!>Ia be of this kind, and you are to find the point
Ky where the projeAed body falls upon any right line AN*
paflihg through the point A, let AN- produced meet the
lAfymptotes MX, NX in M a^ N, and take NK equal to
AM.
* Rule 7. And hence appears iln expeditious method of de-
termining this hyperbola from the phsenomena. Let two iV*
milar and equal bodito be prbje€led with the fame velocity,. ior
difierent angles HAK, hAk (P1.£,F^. 6), and kt themfaU vponr
the plane of the horizon iti Kand k ; and note the proportion
of AK to Ak. Let it be as d to e* Then ev^ing a perpeiL*<
dicular AI of any length, affume any how the length AH ot '
Ail, and thenc6 graphically, or by fcale and compafe, coIle<5l
the lengths AK, Ak <by rule^Q), If tiit ratio^of AK: to Ak bQ
the feme with that of d to e, the .lengtji of: AH w;^ rightly
affusnted. lfno% tate on the indjefinite tight liW)l&M!(iPJ; 2, Fig^
7), the length SM equal to the affww^d AH^.tindnffiSi a perr
pendicular MN equal to the difference -n- ~ - of*ifie ratios
* . • ■ .;.•.... Ak ,.e •)r.;:- . t
drawn into any given right line. By the like m^tiiod, from
feveral affumed lengths AH, you may find feveral. points; N ;
and draw through them all a regular curve NNXN, -cutting
the right line SMMM in X. Laftly, affume AH eqmal to the
ahfciffa SX, aijd thence find ^dn the length AK; aod the
lengths, which ai'e to the affumed length AI, and this laft AH*
as the length AK known by experiment, to tfce lengtii AK
laft found, will be the true lengths AI and AH, whicii were
to be found. But thefe being given, there will be given: aJfo
the refilling forpe o£ the medium in the place A, it being to
SeS. II. OP NATURAL :i^HlLOSOPHY. ' S3
the force of gravity as AH to SAI, Let the denfity of the
medium be increafed by rule 4, and if -the refilling force juft
found be inci^afed in the fame ratio^ it will become flill
more accurate.
Rule 8. The lengths AH, HX being found ; let there be
BOW required the pofitif^ of the line AH, according to which
a projeAile thrown with that given velocity (hall fall upon any
point K. At the points A and K (PI. 2, Fig. 6) ere6l th^ lines
AC, KF perpendicular to the horizon ; whereof let AC be
drawn downwards, and be equal to AI or i^HX. With the
afymptotes AK, KFi defcribe an hyperbola, whofe conjugate
fliall pafs through the point C ; and from the centre A, with
(he interval AH, defcribe a circle cutting that hyperbola in
the point H ; then the projedlile thrown in the diredion of
the right line AH will fall upon the point K. Q.E.I. For
the point H, becaufe of the given length AH, muft be fome-
where in the circumference of the defcribed circle* Draw
CH meeting AK and KF in E and F ; and becaufe CH, MX
are parallel, and AC, A I equal, AE will be equal to AM,
and therefore alfo equal to KN. But CE is to AE as FH to
KN, and. therefore CE and FH are equal. Therefore the
point H falls upon the hyperbolic curve defcribed with the
afymptotes . AK, KF, whofe conjugate pafles through the
point C ; and is therefore found in the common interfeftion
of this hyperbolic curve and the circumference of the defcribed
circle. Q.E.D. It is to be obferved that this operation is
the fame, whether the right line AKN be parallel to the ho-
rizon, or inclined thereto in any angle ; and that from two
interfe<ftions H, H,- there arife two angles NAH, NAH ; and
that in mechanical pra6lice it is fufEcient once to defcribe a
circle, then to apply a ruler CH, of an indeterminate length,
foto the point C, that its part FH, intercepted between the
circle hud the right line FK, may be equal to its part CE placed
between the point C and the right line AK.
What has been faid of hyperbolas may be eafily applied to
parabolas. For if (PI. 2, Fig. 8) a parabola be reprefented by
XAGK, touchedby a right line XV. in the vertex X, and the
ordioaies lA, VG be as any powers XP, XV% of the ab-
VoL. 11. D
M MAXTHBMATIOAL PmiNt^IPlM Book IL
kWmXl, XV; draw XT, GT, AH, whei«of let XT be pa*
mUel to VG, and let <3T, AH touch the parahola in G and
A : and a body prq^edted firom any place A, in the direSioa
of the right line AH, with a due velocity, will defcribe this
parabola, if the deafity of rti^ medium in eaoh of the places
,G he redpcocally as the tangent 6T« j^ la that cafe the velo-
city in G wilt be the fame as would ootfe a body, moving in
a nonHrefiiling fpace, to defcribe a conic parabola, having O
for its vertex, VG produced downwards for its diameter, and
2Gy
mi— n X VG
for its latus retShmi. And the jefifling forca
Q||T| • ir I ^n
in G will be to the force of gravity as GT to VG«
Therefore if NAK reprefent an horizontal line, and both the
denfity of the medium at A, and the velocity with which the
' body is proje6led, remaining the fame, the angle NAH be
any how altered, the lengths AH, AI, HX will remain ; and
tiience wiH be given the vertex X of the parabola, and the
pofition of the right line XI ; and by taking VG to lA as XV*
to XP, there will be given all the points 6 of the parabola^ *
Arough which the projeftile -will pafs.
SECTION HI.
* Of the motions of bodies which are refijied ptertiy in the ratio
of the velocities, and partly in the duplicate of the famt
ratio.
TROPOSmON XI. THEOREM VIII.
Jf a tody be refifitd partly in the ratio andpixrtly in the A^
plicate ratio of its velocity, and moves in afimilar medium
by its innate force only ; and the times be taken in antk"
meticdl progrejjion; then quantities reeiprocaUy propot^
tional to the velocities, increafed by a certain given quam-
tity, will be in geometf'ical progrejjion. (PI. 3, Fig. i.)
With the centre C, and the redtanguhu: afymptotes CADd
and CH, defcribe an hyperbola BEe, and let AB, DE, de, be
parallel tothi^afymptote CH. In the afymptote CD let A, 6
be given points ; and if the time be expounded by the hyper-
bolic area ABED uniformly increafing, I fay, that the ydo^
<iXj may be exprefled by the length DF, whofe lec^rooal
j%*n.fs/ii.
lOffSoi^WaiU^aJr S£r*£i.
■*■
^J)f togetfi^wijth the given l^ue CG, copipofethelqngtb CD
increafing in a geometrical progfeffion.
For }et the arepla DEed be the lea^t g^v^ i^u^rement of Ihe
iime^ ai^d !Pd will be reciprocally as DE, and thereifore di«
1
reftly as CD. Theref<p|re the decrement of ^^j^J which (hjr
lem. 2, book £) is -^^^^ will be alfo as ^^ or -^^j — ^ that
is^as rTK+ /^rs* Therefore the time ABED mnifijrmly in-
^resifing by the addition of the given pc^rtic^e^ flDde^ it fii^'
lows that ttt^ decreafes in the fame ratio with the velocity.
For the decrement of tbe velocity is as the x^fiflance, t^at is
(by the fuj^ofiiion), ^s the fum ^.two q^antities^ whereof one
is as the velocity, and the other as thefqvane of the velo€;ity ;
and the decrement of ttj^ is as the fum of tl^e .quantities -rrf^
apd TTfy} whereof the firft is -^ itfelf, and the laft ttt^
1 1 .
is asTTTT,; therefore ^^ is as the velocity, the decrements
of bodi being aoak^us. And if <lhe quantity GD, recipro^
cally proportional to ^^j be augmented by the given quan-
tity CO; tiie fum' CD, ihe tiine AB&D uniformly inci»afing,
will increafe in a geometrical progreffion. Q.E.D. *
CoR. 1. Therefore, if, having the points A and G given,
the time be expounded by ih^ hyperbolic area ABED, the ve-
1
locitymay be expounded by jr^ the reciprpqal of GD.
Cor. 2. And by taking GA to GD ^ ihft reciprocal 4>f the
•velocity at the beginning to the recifwocai of the velocity at
the end of any time ABED, the point G will be ^und. And
that point being found, the velocity may be found from any
other time given.
PROPOSITION XII. THEOREM IX.
The fame things being fuppofed, I fay, that if thefpacesde-
fcribed are taken in arithmetical progreffion, the velocities
D S
36 MATHEMATICAL PRINCIPLES Book if.
augmented by a certain given qtiantity will be in geometrical
progrejfion. (PI. 3, Fig. 2.)
In the afymptote CD let there be given the pbint R, and,
ereAing the perpendicular RS meeting the hyperbola ift S, let
the fpace defcribed be expounded by the hyperbolic area
RSED; and the velocity will be as the length GD, which,
together with the given line CG, compofes a length CD de-
creafing in a geometrical progreffion, while the fpace RSED
increafes in an arithmetical progreflion.
For, becaufe the increment EDde of the fpace is given,
the lineolsB Dd, which is the decrement of GD, will be reci-
procally as ED, and therefore dire6lly as CD ; that is, as the
fum of the fame GD and the given length CG. But the de-
crement of the velocity, in a time reciprocally proportional
thereto, in which the given particle of fpace DdeE is de-
fcribed, is as the refiftance and the time conjun6):ly,-that is,
diredily as the fum of two quantities, whereof one is as the
velocity, the other as the fquare of the velocity, and inverfely
as the velocity ; and therefore dire6lly as the fum of two
quantities, one of which is given, the other is as the velocity.
Therefore the decrement both of the velocity and the line GD
is as a given quantity and a decreafing quantity conjundlly ;
and, becaufe the decrements are analogous, the decreafing
quantities will always be analogous ; viz. the velocity, and the
l^neGD. Q.E.D.
CoB. 1 . If the velocity be expounded by the length GD, the
fpace dfefcribed Will be as the hyperbolic area DESR.
Cob- 2. And if the point R be aiTumed any how, the
point G will be found, by taking GR to GD as the velocity
at; the beginning to the velocity after any fpace RSED is de-
fcribed. The point G being given, the fpace is given from
the given velocity : and the contrary.
Cob. 3. Whence fince (by prop. 11) the velocity is given
from the given time, and (by this prop.) the fpace is given,
from the given velocity ; the fpace will be given from the
given time : and the contrary.
SeSi. III. OF NATURAL PHILOSOPHY. 37
PROPOSITION XIII. THEOREM X.
Suppojing that a body attraded downwards by an uniform
gravity afcends or defcends in a right line ; and that the
JamdHs rejijied partly in the ratio of its velocity, and partly
in the duplicate ratio thereof: I fay, that, if right lines pa-
rallel to the diameters of a circle and an hyperbola be drawn
through the ends of the conjugate diameters, and the velo»
cities be asfome figments of thofe parallels dratsmfrom a
given pointy the times will be as thefeSlors of the areas cut
off by right lines drawn from the centre to the ends of the
fegments ; and th^ contrary. (PI. 3, Fig. 3.)
Case l. Suppofe fii*ft that the body is afcending, and from
the centre D, with any femi-diameter DB, defcribe a qua-
drant BETF of a circle, and through the end B of the femi^
diameter DB draw the indefinite line BAP, parallel to the fe-
mi-diameter DF, In that line let there be given the point A,
and take the fegmcnt AP proportional to the velocity. And
fince one part of the refillance is as the velocity, and another
part as the fquare of the velocity, let the whole refiftance be
as AP* + 2BAP. Join J) A, DP, cutting the circle in E and T,
and let the gravity be expounded by DA% fo that the gravity
fhall be to the refiftance m P as DA* to AP* 4. 2BAP ; an4
the time of the whole afcent will b^ as the fedlor EDT of the
circle.
For draw DVQ, cutting off the moment PQ of the velocity
AP-, and the moment DTV of the fecftor DET anfweringto a
given moment of time ; and that decrement PQ 6f the velo-
city will be as the fum of the forces of gravity DA* and of re-
fiftance AP* + 2BAP, that is (by J2 prop. 2 book, elem.), as
DP*. Then the area DPQ, which is proportional to PQ, is
as DP*, and the area DTV, which is to the area DPQ as DT*
to DP*, is as the. given quantity DT*. Therefore the arei^
EDT decreafes uniformly according to the rate of the future
time, by fubdu6iion of given particles DTV, and is therefore
proportional to the time of the whole afcent. Q.E.D.
Case 2. If the velocity in the afcent of the body be ex
. pounded by the length AP as before, and the refiftance be
made as AP* + 2BAP, and if the force of gravity be lefs
1)3
S8 rixtflfitirfATicAL i?!iiNcii?LES Book If.
than can be expreffed by DA*; take BD (Fig. 4) of fuch a
length, that AB* — BD* may be proportional to the gravity,
and let DV be perpendicular and equal to DB, and through
the vertex F defcribe the hyperbola FTVE, whofe conjugate
fertii-diameters ate DB and DF, and which cuts DA in E,
end Dp, DQ in T and V ; and the time of the whole afcent
win be as the byperbofic fedor TDE.
For the decrement PQ of the velocity, produced in a given
particle of time, is as the fum of the refiftance AP* + 2BAP
and of the gravity AB* — BD*, that is, as BP* — BD*. But
the area DTV is to the area DFQ as DT* to DP* ; and, there-
fore, if GT be drawn perpendicular to DF, as GT* or GD*
— DF* to BD*, and as GD* to BP*, and, by divifion, as DF*
to BP* — BD*. Therefore fince the area DPQ is as PQ, that
is, as BP* — BD*, the area DTV will be as the given quan*
tity t)F*. TTierefore the area EDT decreafes uniformly in
each of the equal particles of time, by the fubdudlion of fo
many given particles DTV, and therefore is proportional to
the time. Q.E.D.
Case 3. Let AP be the velocity in the defcent of the body,
tod AP* 4- ^BAP the force of refiftance, and BD* — AB* the
force of gravity, the angle DBA being a right one. And if
with the centre D, and the principal vertex B, there be de-
fcribed a redlangular hyperbola BETV (PL 3, Fig. 5) cutting
DA, DP, and DQ produced in E, T, and V ; the feftor DET
of this hyperbola will be as the whole time of defcent.
For the increment PQ of the velocity, and the area DPQ
proportional to it, is as the excefs of the gravity above the
refiftance, that "is, as BD* — AB* — 2BAP — AP* or BD*
-- BP*. And the area DTV is to the area DPQ as DT* to
DP* ; and therefore as GT* or GD* — BD* to BP*, and as
GD* to BD*, and, by divifion, as BD* to BD* — BP*. There-
fore fince the area DPQ is as BD* — BP*, the area DTV
will be as the given quantity BD*. Therefore the area EDT
increafes uniformly in the feveral equal particles of time by
the addition of as many given particles DTV, and therefore is
proportional to the time of the defcent. Q^E.D.
Se&. Ill* <3^ NATV&AL PaiLOSOl^HY.. 3^
Cob. If with the centre D and the femi-diamelia^ DA there
be drawn thnmgh the vertex A an arc At itmihor to the arc
BTy and fivaiajrly fohtending the angle ADT^ the velocity
'AP win be to the velocity which the body in the time IDl^
in a Qou-refifting fpaee^ can lofe in iu afeent^ or acquire in
its defcent^ as the area of the triangle DAP to the area of the
ftdor DAt ; and therefore is given from the time given. For
the velocity in a non-refifling medium is proportional to the
tiine^ and therefore to this, fedtor ;, in a refifting mfidium^ it
is as the tciai^e ; and in both mediums^ where it is leaft^ it
approachesto the ratio of eqiiaKty^ asthefe^kMr md triangle
SCHOUUM.
One may denlonftrate al£b that cafe in the afi:ent of the
body^ where the force of gravity is lefs than caa be expreffed
by DA* or AB* + BD% and greater than can be expreflfed by
AB^ — D&% and moil be expreffed by AB^. But I haflen
to other things.
PKOPOSinON XIV. THEOREM XL
The fame things being fuppofed, Ifay^ that thefpace defcribed
in the afcent or defcent is as the dijference of the area hy
which the time is expreffed^ and offome other area which is
augmenfid or diminished in an arithmetical progrejfitm ; if
the forces compounded of the refijtance and the gravity Be
taken in a geometrical progreffion. (PI. 3, Fig. 6,, 7, B.)
Take AC (in the three laft figures) proportional to the gra-
vity, and AK to the refiftance ; but take them on the fame
fide of the point ^ if the body is defcending, otherwHe on
the contrary. Ere^Ab, which make to DB as DB^ to 4BAC:
and to the re<9:angwlar afymptotes CK, CH, defcribe the hys-
pcrbofei bN ; and, erecSting KN perpendicular to CK, the area
AbNK will be augmented or dhnrniflied in an arithmetical
progreffion, while the forces CK are taken in a geometrical
'jMt>greffion, I fay^ therefore, that the diftance of tlie body from
its. greateft ahitnde is asr the exceis of the area AbNK above
the area DET.
For fince AK is as the riefiilance^ that, is, as AF* x
8BAP; aflume any given quantity Z, and put AK equal to
D 4
40 MATHEMATICAL PRINCIPLES Book II.
— 1^ y then (by lem. 2 of this book) the moment
i^T r AIT 11 u 1 . ^APQ + 2BA xPQ 2BPQ
KL of AK will be equal to y or — = — 9
and the moment KLON of the area AbNK will be equal to
gBPQ X LP BPQ X BI)^
Z ^^2Z X CK >c AB*
Case 1. Now if the body afcends, and the gravity be as
' AB* + BD% BET (in Fig. 6), being a circle, the line AC,
which is proportional to the gravity, will be = — -5 and
DP* or AP* + 2BAP + AB» + BD* will be AK x Z + AC
X Z or CK X Z ; and therefore the area DTV will be to
the area DPQ as DT* or DB* to CK x Z.
Case 2. If the body afcends, and the gravity be as AB*
— BP% the line AC (in Fig. 7) will be ^ ^ andDT*
will be to DP* as DP or DB* to BP* — BD* or AP*+2BAP
+ AB* — BD% that is, to AK x Z + AC x Z or CK x Z.
And therefore the area DTV will be to the area DPQ as DB*
toCKxZ.
Case 3. And by the fame reafoning, if the body defcends,
and therefore the gravity is as BD* — AB% and the line AC
gj)a AB*
(in Fig. 8) becomes equal to ^ J the area DTV
will be to the area DPQ as DB* to CK X Z : as above. .
Since, therefore, thefe areas are always in this ratio^ if for /
the area DTV, by which the inoment m the time, always
equal to itfelf, is expreffed, there be put any determinate re6l
angle, as BD x m, the area DPQ, that is, iBD x PQ, will
be to BD X m as CK x Z to BD\ And thence PQ x BD*
becomes equal to 2BD x m X CK x Z, and the moment
KLON of the area AbNK, found before, becomes
BP X BD X m _ , _ _^ n , 1 o .
.p • rrom the area DET fubduct its moment
AU
DTV or BD X m, and there will remain ar ' • 1^^?^
Sf6l. III. OF NATURAL PHILOSOPHY. 41
fore the difference of the moments, that is, the moment of
,1/^ /.I , APxBD X m
the difference or the areas, is equal to j» . and
therefore (becaufe of the given quantity -t^ — J as the ve-
locity AP ; that is, as the moment of the fpace which the
body defcribes in its afcent or defcent. And therefore the
difference of the areas, ?ind that fpace, increafing ordecreaf-
ing by proportional moments, and beginning together or
vanifiiing together, are proportional. Q.E.D.
Cob. If the length, which arifes by applying the area DET
to the line BD, be called M ; and another length V be taken
in that ratio to the length M, which the line DA has to the"
line DE ; the fpace which a body, in a refifting medium, de-
fcribes in its whole afcent or defcent, will be to the fpace
which a body, in a noh-refifting medium, falling from reft, can
defcribe in the fame time, as the difference of the aforefaid
BD X V*
areas to -t^ — J and therefore is given from the time
given. For the fpace in a non-refifting medium isin aduplicate
ratio of the time, or as V*; and, becaufe BD and AB are given,
BD X V* _, . . , ^ DA* X BD X M*
as .p • This area is equal to the area — rr-pr — rr=; —
AB ^ - , D£*xAB
and the moment of M is m; and therefore the moment of
,. . DA*xBDx2Mxm -„ ,. . ^
this area is jrpi — j-^ • But this moment is to the
moment of the difference of the aforefaid areas DET and
Ai^xTt^ • . Al^xBDxm DA^xBDxM ,„^ .
AbNK, VIZ. to -^ ^ } as ^^ to ^BD x
DA*
AP, or as ^^^ into DET to DAP ; and, therefore, when the
areas DET and DAP are leaft, in the ratio of equalitji. There-
BD X V*
fore the area — -^ — and the difference of the areas DET
and AbNK, when all thefe areas are leaft, have equal moments;
• und are therefore equal. Therefore fince th^ velocities, and
4^ MATHEMATICAL PKINCIPLBS ffook 11^
tJberefore alfo'tbe fpaces in both imediuma defcribed together^
in the beginning of the defcent, or the end of the afcent, ap-
proach to eqiiiKty, and therefore are then one to another as^
BD X V*
the area .^ — > and the difference of the ar^as DET and
AbNK^ and moreover fince the fpace^ in a non-refifting me-
BD X V*
dium, is perpetually as ■ .p — j and the fpace, in a refift-
mg medium^ is perpetually as the diiference of the areas DET:
and AbNK; it neceiTarily follows^ that the fpaces^ in both
tiiediums> defcribed in any equal times^ are one to another
BD X V*
as that area — -^ — 9 and the difference of the areas DET
and AbNK, Q.E.D.
SCHOLIUM.
The refiftance of fpherical bodies in fluids arifes partly
Krom the*tenacity^ partly from the attrition; and partly from^
the deniit^ of the medium. And that part of the trefiftance
which ar|fes from the denfity of the fluids is> as I faid^ in a
duplicate ratio of the Telocity ; the other part, which arifes
frcnen the tenacity of the fluids is uniform^ or as the moment
of the time; and, therefore, we might now proceed to the
Blotion of bod^, which are refifled partly by an uniform
:KMrce, or in the ratio of the moments of the tinie> and partly
in the duplicate ratio of the vdlocity. But it is fufEcient to
have cleared the way to this fpeculation in the 8th and Qth
l^rop. feiregoing, and their corollaries. For in Ihoie propofitions,
infiead of ^heiiniform refiftance made toai afcending body
ariiing from its gravity, one may fubftitute the uniform refift»
ance which arifes from the tenacity of the medium, when the
body moves by its vis infita alone; and when the body
aleends in^. a right line, add this uniform refiftance to the
force of gravity, and fubdu6); it when the body defcends in a
right line. One might alfo go on to the motion of bodies
which are refifted in part uniformly, in part in tb(e ratip of the
telocity, and in part in the dupUc$tte vatio of the feme velo-
fiaum.iwa
#
» »
■*
'♦
Se^. m. o* Natural PHiLosoi^ftt. 45
city. And I haVe opened a way to this in the 13th and 14th
prop, foregoing^ in which the uniform refiftance drifing from
the tenacity of the medium riiay be fubftituted fbr the force
of gravity, or be compounded with it as before. But I haften
to other things.
SECTION IV.
Of the circular motion of bodies in refifiing me^ms.
LEMMA III.
Let PQR be afpiral cutting all the radii SP, SQ, SR, S;e, in
equal angles. Draw the right line FT touching the fpiral
in any point P, and cutting the radius SQ in T ; draw PO,
QO perpendicular to the fpiral , and meefing in O, and join
SO. I fay 9 that if the points P and Q approach and coin--
cide, the angle PSO will become a right angle, and the ulti^
mate ratio cf the reSangle TQ X 2PS to PQ* will be the
ratio of equality. (PI. 4, Fig. I.)
For from the right angles OPQ, OQR, fubdu6l the equal
angles SPQ, SQR, and there will remain the equal angles
OPS, OQS. Therefore a circle which paffes through the
points OSP will pafs alfo through the point Q. Let the
points P and Q coincide, and this circle wilt touch the fpiral
in the place of coincidence PQ, and will therefore cut the
tight line OP perpendicularly. Therefore OP will become
a diameter of this circle, and the angle OSP, being in a femt«
circle, becomes a right one. Q.E.D.
Draw QD, SE perpendicular to OP, and the ultimate ratios
of the lines will be as follows: TQ to PD as TS or PS to PE,
or 2PO to 2PS; and PD to PQ as PQ to 2PO ; and, ex aqua
perturbatiy TQ to PQ as PQ to SPS. Whence PQ* becomes
equal to TQ X 2PS. Q.E.D.
PROPOSITION XY. THEOREM XII.
If the denfity of a medium in each place thereof be reciprocally
as the diftance of the places from an immovable dentre, and
the centripetal force be in the duplicate ratio of the denfity ;
I f^yy that a body may re^ohe in a fpiral which cuts all
the radii drawn from that centre in a given angle. (PL 4,
Kg. S.)
44 MATHEMATICAL PRINCIPLES Book II
Suppofe ever)' thing to be as in the foregoing lemtna^ and
produce SQ to V fo that SV may be equal to SP. In any
time let a body^ in a refiiliiig medium^ defcribe the lead arc
PQ^ and in double the time the lead arc PR; and the
decrements of thofe arcs arifing from the refiftance, or
their differences^ from the arcs which would be defcribed
in a non-refifling medium in the fame times^ will be to
each other as the fcjuares of the times in which they are ge-
nerated ; therefore the decrement of the arc PQ is the fourth
part of the decrement of the arc PR. Whence alfo if the
area QSr be taken equal to the area PSQ, the decrement of
the arc PQ will be equal to half the lineolas Rr ; and therefore
the force of refiftancc and the centripetal force are to each
other as the lineolaes 4Rr and TQ which they generate in the
. fame time. Becaufe the centripetal force with which the
body is urged in P is reciprocally as SP% and (by lem, 10,
* book 1) the lineolae TQ, which is generated by that force, is in
a ratio compounded of the ratio of this force and the dupUr-
cate ratio of the time in which the arc PQ is defcribed (for in
this cafe I negled; the refiftance, as being infinitely lefs than
the centripetal force), it follows that TQ X SP% that is (by
the lafl; lemma), iPQ* X SP, will be in a duplicate ratio of
4()ie time, and therefore the time is as PQ x v'SP ; and the
Velocity of the body, with which the arc PQ is defcribed in
PQ 1 . .
that time, as r-— rfpOr — ^^5 that is, in the fubduplicate
ratio of SP reciprocally. And, by a like reafoning, the velo-
city with which the arc QR is defcribed, is in the fubduphcat^
ratio of SQ reciprocally. Now thofe arcs PQ and QR are a^
the defcribing velocities to each other ; that is, in the fubdu-
plicate ratio of SQ to SP, or as SQ to ^SPxSQ; and, be-
caufe of the equal angles SPQ, SQr, and the equal areas PSQ,
QSr, the arc PQ is to the arc Qr as SQ to SP. Take the dif-
ferences of the proportional confequents, and the arc PQ will
be to the arc Rr as SQ to SP — ^SPxSQ, or ^VQ. For
the points P and Q coinciding, the ultimate ratio of SP
— ^SPxSQ to ^VQ is the ratio of equality. Becaufe the
decrement of the arc PQ arifing from the refiftance, or its
•
SeS. IV. OF NATURAL PHILOSOPHY. 45
double Rr, . is as the reliftapce and the fquare of the time
Rr ,
conjun6Uy, the refiftance will be as poTTTsP* BulPQ was
Rr
to Rr as SQ to |VQ, and thence p^^^^ becomes as
|VQ iOS ^ . . ^.
POTSP x"SQ'^' ^ OPTSP*' ^^' '^^ P^*"'^ P. ^°^ Q'
coinciding, SP and SQ coincide alfo, and the angle PVQ be-
comes a right one ; and^becaufe of thq iimilar triangles PVQ,
PSO, PQ becomes to |VQ as OP to iOS. Therefor*
OS ■ ' ■ ■ ! . ■•.•',:
T^rr — rrfTi is as the refiftance,. that is, in the ratio of the dea-
Ur X bJr* ■
fity of the medidm in P and the duplicate ratio of the velociw
ty conjunctly • SubducSl the duplicate ratio of the velocity,
namely^ the ratio* nrjj and there will remain the denfity of
OS
the i^edium in P^ as jfTpTTcp* ^^ , the fpiral be given,
and, becaufe of the given ratio of OS to OP, the denfity of
1
the medium in P will be as ^» Therefore jn a medium whofe
denfity is reciprocally as SP the diftance from the centre, 4
body will revolve in this fpiral. Q.E.D.
Cob. !• The velocity in any place P, is always the fame
wherewith a body in a non-refifting medium with the fame
centripetal force would revolve in a circle, at the fame
diftance SP from the centre.
CoR. 2. The denfity of the medium, if the diftance SP be
OS .... OS
given, is as ----> butifthatdiftanceisnotffiven,as-5:r •
^ OP' ^ OPxSP
And thence a fpiral may be fitted to any denfity of the pie-
dium.
CoR. 3. The force of the refiftance in any place P is to
the centripetal force in the fame place as lOS to OP. For
thofe forces are to each other as |Rr and TQ, or as — — --
I
44 MATHEMATICAL PRINCIPLES Sook II
Suppofe ever}' thing to be as in the foregoing lemma^ and
produce SQ to V fo that SV may be equal to SP. In any
time let a body^ in a reiiiliiig medium^ defcribe the lead arc
PQ^ and in double the time the lead arc PR; and the
decrements of thofe arcs ariiing from the refiftaiice^ or
their differences from the arcs which would be defcribed
in a non-refifling medium in the fame times^ will be to
each other as the fcjuares of the times in which they are ge-
nerated ; therefore the decrement of the arc PQ is the fourth
part of the decrement of the arc PR. Whence aUb if the
area QSr be taken equal to the area PSQ, the decrement of
the arc PQ will be equal to half the lineola^ Rr ; and therefore
the force of refiftancc and the centripetal force are to each
other as the lineolaes 4Rr and TQ which they generate in the
. fame time. Becaufe the centripetal force with which the
body is urged in P is reciprocally as SP*, and (by lem. 10,
" book 1) the lineolae TQ, which is generated by that force, is in
a ratio compounded of the ratio of this force and the dapUr
cate ratio of the time in which the arc PQ is defcribed (for in
this cafe I negle6l the refiftance, as being infinitely lefs than
the centripetal force), it follows that TQ X SP% (hat is (by
the laft lemma), iPQ* X SP, will be in a duplicate ratio pf
•%lhe time, and therefore the time is as PQ X v'SP ; and the
velocity of the body, with which the arc PQ is defcribed in
PQ 1 . .
that time, as r—- TcoOr ^p 9 that is, in the fubduplicate
ratio of SP reciprocally. And, by a like reafoning, the velo-
city with which the arc QR is defcribed, is in the fubduplicate
ratio of SQ reciprocally. Now thofe arcs PQ and QR are a^
the defcribing velocities to each other ; that is, in the fubdur
plicate ratio of SQ to SP, or as SQ to ^SPxSQ; and, be-
caufe of the equal angles SPQ, SQr, and the equal areas PSQ,
QSr, the arc PQ is to the arc Qr as SQ to SP. Take the dif-
ferences of the proportional confequents, and the arc PQ will
be to the arc Rr as SQ to SP — •SPxSQ, or iVQ. For
the points P and Q coinciding, the ultimate ratio of SP
— '^SPxSQ to 4VQ is the ratio of equality. Becaufe the
decrement of the arc PQ arifing from the refiftance, or its
•
SeS. IV. OF NATURAL PHILOSOPHY. 45
double Rr, , is as the refiftapce and the fquare of the Lime
Rr
conjun6Uy, the refiftance will be as poTTTsP* ButPQ was
Rr
to Rr as SQ to |VQ, and thence -^rr^ — ^ becomes as
PQITSP^SQ^^'^OPTsF" For thepomtsP.and Q.
coinciding, SP and SQ coincide alfo, and the angle PVQ be-
comes a right one ; and, becaufe of the iimilar triangles PVQ,
PSO, PQ becomes to fVQ as OP to iOS. Therefor*
OS
TTTT — rrfr: is Bs the refiftance,, that is, in the ratio of the dea-
OP xbP* ^
fity of the medium in P and the duplicate ratio of the veloci^
ty conjunctly. SubducSl the duplicate ratio of the velocity,
• ' • 1
namely^ the ratio np5 and there will remain the denfity of
OS
the i^edium in P, as jrypTTcp* Le*^ . ^^^ fpiral be given,
and, becaufe of the given ratio of OS to OP, the denfity of
the medium in P will be as '^* Therefore in a medium whofe
denfity is reciprocally as SP the diftance from the centre, ^
body will revolve in this fpiral. Q.E.D.
Cob. 1. The velocity in any place P, is always the fame
wherewith a body in a non-refifting medium with the fame
centripetal force would revolve in a circle, at the fame
diftance SP from the centre.
CoR. 2. The denfity of the medium, if the diftance SP be
OS OS
given, is as r=—-, but if that diftance is not given, as
OP' ° ' OPxSP
And thence a fpiral may be fitted to any denfity of the pie-
dium.
CoR. 3. The force of the refiftance in any place P is to
the centripetal force in the fame place as 4OS to OP. For
IVQ X PQ
thofe forces are to each other as |Rr and TQ, or as2 — --- — !_:
oQ
46 MATHEMATICAL PRINClPLEl Book II*
and -i^^, that is, as iVQ and PQ, or ^OS and OP. The
Ox
fpiral, therefore, being given, there is given the proportion
of the r^fiftance to the centripetal force ; and, vice vtrfa,
from that proportion given the fpiral is given.
CoR. 4. Therefore the body cannot revolve in this fpiral^
except where the force of reiiftance is lefs than half the oenr
iripetal force. Let the reiiftance be made equal to half
jthe centripetal force, and the fpiral will coincide with
^ right line PS, and in that right line the body will
defcend to the centre with a velocity that is to the velo-
city, with which it was proved before, in the cafe of the para^
bola (theor. 10, book 1), the defcent would be made in »
|ion-re&fiing medium, in the fubduplicate ratio of unity to
the number two. And the times of the defcent will be here
reciprocally as the velocities, and therefore given.
CoR. 5. And becaufe at equal diftances from the centre tljfC
Telocity is the fame in the fpiral PQR as it is in the right
line SP, and the length of the fpiral is to the length of tbe
light line PS in a given ratio, namely, in the ratio of OP \o
OS ; the time of the defcent in the fpiral will be to the time
lof the defcent in the r^ht line SP in the fame given ratio^ mi
therefore given.
CoR. 6. If from the centre S, with any two givep iaJervals^
two circles are defcribed ; and thefe circles remaining, tjh^
angle which the fpiral makes with the radiu3 PS be Wf
how changed ; the number of revolutions which the bQdjT
can complete in the fpace between the circumfereAces of
thofe circles, going round in the fpiral fK)m one circum-
PS
fetence to another, will be as ■ ■, or as the tangent of the
angle which the fpiral makes with the radius PS; and rtb^
OP
time of the fame revolutions will be as -tt^, that is^ as the
(Tecant of the fame ai^le, or jeciprocally as the denfity of the
medium.
GoR. 7. If a body, in a medium whofe denfity is recipro^
cally as the diftances of places from the centre, revolves in any
ScS. IT. t)F NATURAL PHILOSOPHY. 47
curve AEB (P1.4,Fig, 3) about that centre, and cuts thefirftradiu*
AS in the iame angle in B as it did before ia A, and that with
a velocity thatihall be to its firft velocity in A reciprocally in
a fubduplicate ratio of the diftances' from the centre (that i^
as AS to a mean proportional between AS and BS) that body
will continue to defcribe innumerable fimilar revolutions BFQ,
OGD, &c. and by its interfedions will diflinguifli the radius
AS into parts AS^ BS, CS, JDS, &c. that are continually pro-
|rortioDal. But the times of the revolutions will be as the
perimeters of the orbite AEB, BFC, OGD, &c. dire<aiy, aa4
the velocities at the beginnings A, B, C of tho& orbits inverfo-
ly ; that is, as AS* BS*, CS*. Ar^d the whole time in
which the body will arrive at the centre, will be to the time
of the firft revolution as the fum of all the continued propor-
tionals AS^^ BS% CS^, going on ad infinitum, to the firft
teixn AS*; tfiat is, as the firft term AS* to the difference of
the two firft AS^ — BS , or as f AS to AB very pearly*
Whence the whole time may be eafily found.
Cob. 8. From hence alfo may be deduced, near enough,
the motions of bodies in mediums whofe denfity is either uni-
form, or obferves any other affigned law. From the centre
S, with intervals SA, SB, SC, &c. continually proportional,
defcribe as many circles; and fuppofe the time of the revolu-
tions Jbetween the perimeters of any two of thofe circles, in
the medium whereof we treated, to be to the time of the revo-
lutions between the fame in the medium propofed as the mean
denfity of the propofed medium between thofe circles to the
mean denfity of the medium whereof we treated, between the
fame circles, nearly : and that the fecant of the angle in
which the fpiral above determined, in the medium whereot
we treated, cuts the radius AS, is in the fame ratio to the
fecant of the angle in which the new fpiral, in the propofed
medium, cuts the fame radius: and alfo ths^ the number of
all the revolutions between the fame two circles is nearly as
the tangents oftBofe angles. If this be done every w^ere
48 MATHEMATICAL PRrKrU>LES BooJk If •
between every two circles^ the motion will be continaed
^through all the circles. And by this means one may without
difficulty conceive at what rate and in what time bodies ought
to revolve in any regular medium.
Cor. g. And although thefe motions becoming eccentrical
fhould be performed in fpirals approaching to an oval figure^
yet, conceiving the feveral revolutions of thofe fpirals to be
at the fame diftances from each other, and to approach to the
centre by the fame degrees as the fpiral above defcribed, we
may alfo undcrftand how the motions of bodies may be per^
formed in fpirals of that kind.
PROPOSITION XVI. THEOREM XIII.
If the denjitj/ of the medium in each of the places be reciprocally
as the dijtance of the places from tlie immovable centre, and
the centripetal force be reciprocally as any power of the fame
dijlance, I fay, that the body may revolve in a fpiral inter^
feBing all the radii drawn from that centre in a given atiglc*
(Pr.4,Fig.2.)
This is demonftrated in the fame manner as the foregoing
propofition. For if the centripetal force in P be reciprocally
as any power SP"+' of the diftance SP whofe index is
n + 1 ; it will be colle6led, as above, that the time in which
r
the body defcribes any arc PQ, will be as PQ x PS^" ; and the
refiftance in P as =—-- -—-, or as 1 2" X VQ ,
PQv X SP- PQ X SP'HrSQ' ^^'^
therefore as l — J" X OS that is, (becaufe J — jn x OS
OPxSP« + ^ OP
is a given quantity), reciprocally as SP"+'. And therefore,
fince the velocity is reciprocally as SP^", the denfity in P will
be reciprocally as SP.
Cor. 1. The refiftance is to the centripetal force as
1— l-n X OS to OP.
Cor. 2, If the centripetal force be reciprocally as SP', 1 —
^n will be = ; and therefore the refiftance and denfity of
ithe medium will be nothing, as in prop. 9, book 1.
Cor. 3. If the centripetal force be reciprocally as any
power of the radius SP, whofe index is greater than the num-
-OK
Se6t. IV. OF NATURAL PHILOSOPHY. 49
ber 3, the affirmative refiftance will be changed into a nega-
, tive,
SCHOLIUM.
This propofition and the former, which relate to mediums
of unequal denfity, are to be undei-ftood of the motion of
bodies that are fo fmall, that the greater denfity of the medi-
um on one fide of the body above that on the other is not
to be cohfidered. I fuppofe alfo the refiftance, cateris paribus,
to be proportional to its denfity. Whence, in mediums whofe
force of refiftance is not as the denfity, the denfity muft be
fo much augmented or dimihifhed, that either the excefs of
the refiftance may be taken away, or the defedJ; fupplied.
PROPOSITION XVII. PROBLEM IV.
To find the centripetal force and the rejijiing force of the
medium^ by which a body, the law of the velocity being
g}ven,JhaU revolve in a given fpiral. (PI. 4. Fig. 4.)
Let that fpiral be PQR. From the velocity, with which
the body goes over the very fmall arc PQ, the time will be
given; and from the altitude TQ, which, is as the centripe-
tal force, and the fquare of the time, that force will be given.
Then from the diiference RSr of the areas PSQ and QSR
defcribed in equal particles of time, the retardation of the
body will be given; and from the retardation will be found
the refifting force and denfity of the medium.
PROPOSITION XVIII. PROBLEM V.
The law of centripetal force being given, to find the denfity of
the medium in each of the places thereof, by which a body
may defcribe a given fpiral.
From the centripetal force the velocity in each place muft
be found; then from the retardation of the velocity the d^nfitj
of the medium is found, as in the foregoing propofition.
But I have explained the method of managing thefe pro-
blems in the tenth propofition and fecond lemms^ of this book;
and will no longer detain the reader in thefe perplexed dif-
quifitions. I fhall now add fome things relating to the forces
of progreffive bodies, aad to the denfity and refiftance of thofe
mediums in which the motions hitherto treated of, and thofe
akin to them, are performed.
Vol. II. E
so MATHEMATICAL FRINCIPLKS Book II.
SECTION V.
Of the dcnfity and comprejjion of fluids; and of hydroftaiics.
THE DEFINITION OF A FLUID.
A fluid is any body whofe parts yield to any force impreffed on
it, and, by yielding, are eafily moved among themfehes.
PROPOSITION XIX. THEOREM XIV.
Jtll the parts of a homogeneous and tinmoved fluid included in
any unmoved vejfel, and compreffed on every fide (fetting
afide the conjideration ofcondenfation, gravity, and all cen-^
tripetal forces) y will be equally preffed on every fide, and
remain in iheir places without any motion arifing from thai
preffure. (PL 4, Fig. 5.)
Case 1. Let a fluid be included in the fpherical veflel
ABC^ and uniformly comprefled on every fide: I fay, that no
part of it will be moved by that prelTure. For if any part, as
D, be moved, all fuch parts at the fame diftance from the
centre on every fide muft neceffarily he moved at the fame
time by a like motion; becaufe the preifure of them all i»
fimilar and equal ; and all other motion is excluded that does
not arife from that preffure. But if thefe parts come all of
them nearer to the centre, the fluid mufl; be condenfed to-
wards the centre, contrary to the fuppofition. If they recede
from it, the fluid mufl; be condenfed towards the circumference;
which is alfo contrary to the fuppofition. Neither can they
move in any one diredlion retaining their diftance from the
cenUe, becaufe, for the fame reafon, they may move in a con-
trary^dirediioD ; but the fame part cannot be moved contrary
ways at the fame time. Therefore no part of the fluid will be
moved from its place. Q.E.D.
Case 2. I fay now, that all the fpherical parts of this
• fluid are equaUy preiifed on every fide. For let EF be a
fpherical part of the fluid ; if this be hot prefled equally on
eveiy fide, augment the lefler preflure till it be preflfed
equally on every fide; and its parts (by cafe 1.) will remain
in their places. But before the increafe of the prefloie,
they Would remain in their places (by cafe 1); and by
StS^ y* OF NATUBAL PHILOSOPHY. 51
the addition of a new preflure they will be moved^ by
the definition of a fluid, from thofe places. Now tjiefe two
coDclofions coQtradi£i; each other. Therefore it w^s falfe to
fay that the fphere EF was not pxefled equally on every fide.
Q.E.D.
Case 3. I fay befidesj that different fpherical parts have
equal preifores. For the contiguous fpherical parts prefs each
otiiei mutually and equally in the point of contadl (by law 3),
But (by cafe 2) they are preffed on every fide with the fame
force. Therefore any two fpherical parts. not contiguous,
fince an intermediate fpherical part can touch both, will be
preffed with the fame forpe. Q.E.D.
Case 4. I fay now, that aU the parts of the fluid are ewery
where prefled eqUaUy. For any two parts may be touched by
fph^ical parts in any points whatever; and there they will
equally prefs thofe fpherical parts (by cafe 3), and are reci-
procally equally preffed by thepoi (by law 3). Q.E.D.
Case 5. Since, therefore, any part GUI of the fluid is in-
doled by the reft of the fluid as in a yeffel, and is equally preff-
ed on every fide ; and alfoits parts equally prefs one another,
and are at reft among themfelves ; it is manifeft that all the
parts of any fluid as £1111, which is preffed equally on every
fide, do prefs each other mutually and equally^ and are at
refi; among themfelves. QJE J).
Case 6. Therefore tf that fluid be included in a veffel of a
yielding fubilance, or that is not rigid, and be not equally
preffed on ever fide, the fame will give way to a ftronger
preffure, by the definition of fluidity.
Case 7* -And therefore, in an inflexible or rigid veffel, a
fluid will not fuflain a flxonger preffure on one fide than on
the other, but will give way to it, and that in a moment of time ;
becaufie the rigid fide of the veflel does not follow the yielding
liquor. But the fluid, by thus yielding, will prefs againft the
oppofite fide, and fo the preflure will tend on every fide to
equality. And becaufe the fluid, as foon as it endeavours to
recede from the part that is moft preffed, is withftood by the
refiftasice of the veflel on the oppofite fide, the preffure will
£ 2
52 MATHEMATICAL PRINCIPLES Book IT.
on every (ide be reduced to equality, in a moment of time,
without any local motion: and from thence the parts of the
fluid (by cafe 5) will prefs each other mutually and equally,
and be at reft among themfelves. Q.E.I).
CoR. Whence neither will a motion of the parts of the fluid
among themfelves be changed by a preflure communicated
to the external fuperficies, except fo fur as either the figure
of the fuperficies may be fomewhere altered, or that all the
parts of the fluid, by prefling one anotlier more intenfely or
remifsly, may Aide with more or lefs difiiculty among them-
felves.
PROPOSITION XX. THEOREM XV.
If all ilie parts of afpherical Jluidy homogeneous at equal dif
tances from the centre, lying on a fpherical concentric
bottom, gravitate towards the centre of the whole, the bottoih
willfuftain the weight of a cylinder, whofe baft is equal to
the fuperficies of the bottom, and whofe altitude is the fame
with that of the incumbent fluid. (PL 4, Fig. 6.)
Let DHM be the fuperficies of the • bottom, and AEI
the upper fuperficies of the fluid. Let the fluid be diftinguiflied
into concentric orbs of equal thicknefs, by the innumerable
fpherical fuperficies BFK, CGL; and conceive the force of
gravity to a6l only in the upper fuperficies of every orb, and
the adlions to be equal on the equal parts of all the fu-
perficies. Therefore the upper fuperficies AE is prefled by
the fingle force of its own gravity, by which all the parts
of the upper orb, and the fecond fuperficies BFK, will (by
prop. 19), according to its meafure, be equally prefled. The
fecond fuperficies BFK is prefled likewife by the force of its
own gravity, which, added to the former force, makes the
preflure double. The third fuperficies CGLk, according to
its meafure, a6led on by this preflure and the force of its
own gravity befides, which makes its preflure triple. And in
like manner the fourth fuperficies receives a quadruple preflure,
the fifth fuperficies a quintuple, and fo on. Therefore the
preflure a6ling on every fuperficies is not as the folid quanti-
ty of the incumbent fluid, but as the number of the orbs
reaching to the upper furfeicQ of the fluid ; and is equal to the
ria/ew.vc^VL.
/iUt/i^ J£
Fcffl._^
I
Fi^S.
jQU*rfncL^e t^yrut4fae» SteieU-
SeB. V. OF NATURAL PHILOSOPHY. 53
gravity of the loweft orb multiplied by the number of orbs :
that is, to the gravity of a folid wHofe ultimate ratio to the
cylinder above-mentioned (when the number of the orbs is in-
creafed and their thicknefs diminifhed, ad infinituniy fo that
the adlion of gravity from the loweft fuperficies to tb^ upper
moft may become continued) is the ratio of equality. There-
fore the loweft fuperficies fuftains the weight of the cylinder
above determined. Q.E.D. And by a like reafoning the
proportion will be evident, where the gravity of the fluid
decreafes in any affigned ratio of the diftance from the cen-
tre, and alfo where the fluid is more rare above and denfer
below. Q.E.D.
Cor. 1. Therefore the bottom is not prefled by the whole
weight of the incumbent fluid, but only fuftains that part of it
which is defcribed in the propofition; the reft of the weight
being fuftained arch wife by the fpherical figure of the fluid.
CoR. 2. The quantity of the preflure is the fame always at
equal diftances from the centre, whether the fuperficies prefl*-
ed be parallel to the horizon, or perpendicular, or oblique;
or whether the fluid, continued upwards from the comprei&d
fuperficies, rifes perpendicularly in a redlilinear dife^on, or
creeps obliquely through crooked cavities and canals, whether
thofe paflages be regular or irregular, wide or narrow. That
the preffure is not altered by any of thefe circumftances, may
be coUeded by applying the demonftration of this theorem
to the feveral cafes of fluids.
Cob. 3. From the fame demonftration it may alfo be
collecSled (by prop. J9), thAt the parts of a .heavy fluid ac-
quire, no motion among themfely^s by the preflure of the
incumbent weight, except that^mption which griffs from
corajfcnfation. , . .
Cpf^. 4. And theirefore if another Jbody of thefam^ fpecific
gravity, incapable of condenfation^ be immerfed lu.^his fluid,
it will acquire no motion by the preflure. of th^ ij^jpumbent
weight : it will neither defcend. ncf . s^fcend, nor. change its
figure. Tf it be fpherical, it will remain fo, notwithflanding
the preflure; if it be fqnare, it will remaip fquare ; and that,
whether it be foft.ar.fluid; whether it fwims freely in. the
E 3
M RAtHEMAtfCAL l^lllNCl^LES Bdok it,
fluids or lies at the bbttemi. For any intemd patt of a fluid
is in the fame ftate with the ftibtnerfed body ; and the cafb
of all fabtnerfed bodies th^t have the fame magnitude^ figune^
and fpecific gravity, is alike. If a fubmerfed body, retaining
]ti$ weight, (hoiiid diiTolve and put od the form of a fluid, this
body, if before it would have afcended, defceiided, or from
any preffure aflbm^e a taew figure> would now likewife afcetid,
dfeibetid> or put oh a heW figure ; and that, becaufe its gravity
dhd th6 other catlfes df its motion remain. But (by cafe S,
j>top. IQ) it wotild now b6 at reft, and retain its figure^
Therefore alfo in the foJrmer cafe.
CoR. 5. Therefore a body that is fpecifically heavier than
a fluid 6oiitigbous to it will dhk ; and that which is fpecifi*
cally lightisr Will afcend, and attain fo much motion i^d
change of figure as that excefs or defcdl of gravity is ablfe to
produce. For that eicefs or defeift is the fame thing as aft
itopiilfe, by whick a body, btherwife in equitibrio with the
parts of th^ fluid, is ai^ed on ; and niay be compared with tb^
excefs or dtefeft of a weight in one of the fcales of a balatic^«
Cbi^. 6. TheiefoHe bodies placed in fluids have a twofoM
gi^vity;tHe one bmeahd abfolule, the other apparent, vu!gar>
and compi^tive. AWbltite gravity is the whole force With
which the body t^hds downwards ; relative and vulgar gravity-
is the excefs bfgriavity with which the body tends downWal^da
ihore than the Ambient fluid. By the fir ft kitid of gravity
the parts of all fluids and bodies gi*avitate in theif proper
plactes; and Ihetefore their weights takeii together compofe
the Weight of the whbte. Fot the whole taken togethar is
heavy, as tbay he iexpcfriehced in Vefleli fiiH 6f liqubr ; and this
wei^t of the Whbleis 'e(j[tial to the weights of all the'piu1!8>
and is therefore compofed of them. By the other kihid irf
gravity bbdies db tfot gravitate 4n their places; that is, *cotai-
piayed With bne another, they db tiot pi'epbhderaite, but^ Wn-
ieririg ohl^ knodier's eirdiiaVburs to defeetid, tetetahi in their
' proper tflkces, as if they Wfre tiot heavy . Thofe thWigs wMch
are ih ttk ait, aiid do tlot ptepofndkate, 'are cotnmbnly Ibdked
on as toot heavy. Thttfe ^Mfch do prepbnidefratci are com-
monly tec^ned heavy, rh a2s tnudh as they are tibt Aifliaitiietl
SeB.y. OF NATURAIi PHILOSOPHY. 65
by the weight of the air. The common weights are nothing
elfe but the excefs of the true weights above^ the weight
of the air. Hence alfo, vulgarly, thofe things are call-
ed light which are kfs heavy, and, by yielding to the
preponderating air, mount upwards. But thefe arebnly com-
paratively light, and not truly fo, becaufe they defcend in va^
cuo. Thus,in water, bodies which, by their greater or lefs gravity,
defcend or afcend, are comparatively and apparently heavy
or light; and their comparative and apparent gravity or levity
is the excefs or defeft by which their true graVitjr either
exceeds the gravity of the water or is exceeded by it. But
thofe things which neither by preponderating defcend, nor^
by yielding to the preponderating fluid, afcend, although by
their true weight they do increafe the weight of the whole^
yet comparatively, and in the fenfe of the vulgar, they do not
gravitate in the water. For thefe cafes are alike demon-
flrated.
CoR. 7. Thefe things which have been demonftrated con-
cerning gravity take place in any other centripetal forces.
Cor. 8. Therefore if the medium in which any body moves
be a<^ed on either by its own gravity, or by any other centri*
petal force, and the body be urged more powerfully by the
fame force; the difference of the forces is that very motive
force, which, in the foregoiQg proportions, I have coniidered
as a centripetal force. But if the body be tdore lightly urged
by that force, the difference of the forces becomes a centri-
fugal force, and is to be oonfidered as fuch. . j,,
Cor. 9. But fince fluids by preffing the included bodies
do not change their external figures,* it appears alfo (by cor^t
prop. 19) that they will not change the fituation of then* in-
ternal parts in relation to one another; and therefore if
animods were immerfed therein, vXiA that all fenfation did
arife from the motion of their parts, the fluid will neither
hurt th^ immerfed bodies, nor excite any fenfation, unlefs fo
far as thofe bodies may be condenfed by the comprefGon*
And the cafe is tbe fame of any fyfiem of bodies encompaffed
with a compreQing fluid. All the' parts of the fyfiem will be
agitated with tho fame motions as as if they were placed in
a vacuum, and would only retain their comparative gravity ;
£ 4
50 MATHEMATICAL PRINCIPLES Book IL
unlefs fo far as the fluid may ibmewbat reiift their motions^ or
be requifite to conglutinate them by compreflion.
PROPOSITION XXI. THEOREM XVI.
Let the det^ity of any Jiuid be proportional to the compreffion^
and its parts be attraded downwards by a centripetal force
reciprocally proportional to the dijlances from the centre:
I fay, that, ifthofe dijlances be taken continually propor'
tional, the denfities of the Jiuid at thejame dijlances will be
alfo continually proportional, (PI. 5, Fig. 1 .)
Let K^ denote the fpherical bottom of the fluid, S the
centre, SA, SB, SC, SD, SE, SF, &c. diftances continually
proportional. Ereft the perpendiculars AH, BI, CK, DL,
EM, FN, 8ic. which fliall be as the denfities of the medium
in the places A^ B, C, D, E, F; and the fpecific gravities in
thofe places will be as -—^-j -^rrj -^^^ &c. or, which is all
Ao rSo Co
one, as -r-s-J -^7=%^ 7=777^ ^^' Suppofe, firft, thefe gravities
AB BC CD
to be uniformly continued from A to B, from B to C, from C
to Dj &c. the decrements in the points B, C, D, &c. being
taken by fleps. And thefe gravities drawn into the altitudes
AB, BC, CD, &c. will give the preflures AH, BI, CK, &c.
by which the bottom ATV is a6led on (by theor. 15).
Therefore the particle A fuilains all the preflures AH, BI, CK,
DL, 8cc. proceeding in irifinitum ; and the particle B fufl;ains
the ppeflures of all but the firfl; AH ; and the particle C all
but the two firfl; AH, BI; and fo on: and therefore the den-
fity AH of the firfl: particle A is to the denfity BI of the fe-
cond particle B as the fum of all AH + BI + CK + DL,
in infinitum, to the fum of all BI + CK + DL, &c. And
BI the denfity of th^ fecond particle B is to CK the denfity
of the third C, as the fum of all BI + CK + DL, &c. to
the fum of all CK + DL, &c. Therefore thefe fums are pro-
portional to their diflferences AH, BI, CK, &c. and therefore
continually proportional (by lem. 1 of this book) ; and there-
fore the differences AH, BI, CK, 8cc. proportional to the
fums, are alfo continually proportional. Wherefore fince the
denfities in the places A, B, C, &c. are as AH, BI, CK^ &c.
Se£t. V. OF NATURAL PHILOSOPHY. 57
they will alfo be continually proportional. Proceed inter-
miffively, ^nd,extBquo, at the diftances SA, SC, SE, continu*
alJy proportional, the denfities AH, CK, EM will be continu-
ally proportionaL And by the fame reafoning, at any dif-
tances SA,SD, SG, continually proportional, the denfities AH,
DL, GO, will be continually proportional. Let now the points
A, B, C, D, E, &c. coincide, fo that the progreffion of the
fpecific gravities from the bottom A to the top of the fluid
may be made continual ; and at any diftances SA, SD, SG,
continually proportional, the denfities AH, DL, GO, being
all along continually proportional, will ftill remain continu-
ally proportionaL Q.E.D.
CoK. Hence if the denfity of the fluid in twoplaces, as Aand
E, be given, its denfity in any other place Q may be colle<Sled.
With the centre S, and the recftangular afymptotes SQ, SX,
defcribe (Fig. 2) an hyperbola cutting the perpendiculars AH^
EM, QT in a, e, and q, as alfo the perpendiculars HX, MY,
TZ, let fall upon the afymptote SX, in h, m, and t. Make
the area YmtZ to the given area YmhX as the given area
EeqQ to the giv^n area EeaA; and the line Zt produced will
cut off the hne QT proportional to the denfity. Eor if
the lines SA, SE, SQ are continually proportional, the areas
EeqQ, EeaA will be equal, and thence thq areas YmtZ,
XhmY, proportional to them, will be alfo equal; and the
lines SX, SY, SZ, that is, AH, EM, QT continually pro-
portional, as they ought to be. And if the lines SA, SE, SQ,
obtain any other order in the feries of continued proportionals,
the lines AH, EM, QT, becaufe of the proportional hyperbo-
lic areas, will obtain the fame order in another ferieS of quan-
titi^es continually proportional.
PROPOSITION XXII. THEOREM XVII.
Let the denfity of any fluid be proportional to the comprejjion,
and its parts be attra&ed downwards by a gravitation recir
procally proportional to thefquares of the dijlancesfiom the
centre : I fay, that, if the diftances be taken in' harmonic
progrejfkm, the denfities of the fluid at thofe diftances will
be in a geometrical progreffion. (PI. 5, Fig. 3.)
Let S denote the centre, and SA, SB, SC, SD, SE, the dif-
tancesingeometrical progreffion. Ered the perpendiculars AH,
I
M MATHEMATICAL PHINCIPLES Book Ih
BI, CK, kc. which (ball be as the dcnfities of the dsid in the
places A, B, C, D, E, &c. and the rj^ecific graTitie* thereof m tbofe
A l-I T\l C* W
places will be as ^^9 ^Tr^j -qp-^? &c. Suppoio thefe gravi-
ties to be oniformly continued, the tirll from A to B, the
fecood from B to C, the third from C to 1>, &c. And thefe
drawn into tlie altitudes AB^ BC> CD^ DE, &c. or, which is
the fame thing, into the dtftances S\, SB, SC, &c. propor-
tional to thole altitudes, will give -^r-j ^^5 -^j &c. Uie exr
pooents of the prcflVires. Therefore fince the dcnfities are as
the fums of thofe preflTures, the differences AH — BI, BI —
CK« Sec. of the denfLties will be as the differences of tbofe
fums r-T-^ TTii9 ^7T> Sec. With the centre S, and the
afymptotes SA, Sx, defcribe any hyperbola, rutting the per-
pendiculars AH, BI, CK, &c. in a, b, c, &c. and the perpen-
diculars Ht, lu, Kw^ let fall upon the afymptote Sx, in h^ i^ k ;
AH
and the differences of the dcnfities tu, uw, &c^ will be as -^v>
BI
-^^ &c. And the rectangles tu x th, uw x ui^ &c. or tp,
AH X th BI X ui ^ , . , „, ^
uq, &c. as o — 9 — co^' o^c« t"^*- *^* ^«* Aa, Bb, &c.
For, by the nature of the hyperbola, SA is to AH or St as th
AH X th
to Aa, and therefore q-r is equal to Aa. And, by a
BI Xtii
like reafoning, — oq— is equal to Bb, &c. But Aa, Bb, Cc,
&c. are continualiy proportional, and therefore proportional
to their differences Aa -— Bb, Bb — Cc, &c., and therefore
the reftangles tp, uq, 8tc. are proportional to thofe differ-
ences; as alfo the fums of the reftangles tp + uq, or tp + uq
+ wr to the fums of the differences Aa — Cc or Aa — Dd.
Suppofe feveral of thefe terms, and the fum of all the differ-
ences, as Aa — Ff, will be proportional to the fiim of all the
rectangles, as zthn. Increafe the number of terras, and di-
sniniih the diflances of the points A, B, C, &c. in hijinitum,
Se£t. V. OF NATURAL PHILOSOPHY. 59
and tfaofe reAangles will become equal to the hyperbolic area
athn, and therefore the difference Aa — Ff is proportional to
this area. Take now any diftances^ as SA^ SD^ SF^ in har-
monic progrefTion, and the differences Aa — Dd, Dd — Ff
will be equal ; and therefore the areas thlx^ xlnz^ proportional
to thofe difierences will be equal among themfelves^ and the
4enfities St^ Sx, Sz^ that is> AH^ Dip FN^ continually pto^
portional. Q.E.D.
CoR. £. Hence if any twodenlities of the fluids as AH and
BI^ be given^ the area thiu^ anfwering to their difference tu,
will be given ; and thence the deniity FN will be found at any
bright SF^ by taking the area thnz to that given area thiu as
the difference Aa — Ff to the difference Aa — Bb..
SCHOLIUM.
By a like reafoning it may be proved, that if the gravity
of the particles of a fiuid be diminilhed in a triplicate ratio
of the diflaiices from the centre; and the reciprocals of
the fquares of the diftances SA, SB, SC, &c. (namely^
SA' SA* SA^\
^S*' SW* SO^ ^^ tiaken in an arithmetical progreffion, the
denfities AH, BI, CK, &c. wiH be in a geometrical progref-
fion. And if the gravity be diminilhed in a quadruplicate ratio
of the diftances, and the reciprocals of the cubes of the dift-
g^4 gy^4 §^4
ances (as ^t^9 oJK"? op,* &c.) be taken in arithmetical pro-
greffion, the denfities AH, BI> CK, 8cc. will be in gedme-
tt-ical progreffion. And fo in infinitum. Again ; if th^ gra-
vity of the particles of the fluid be the fame at all diftaacesy
tad the diftances be iti arithmetical progfeffion, the denfities
will be in a geometrical progreffion, as Dr. HcMey has found.
If tile gravity be as the diftaiice> and the fquares of the is&j*
ances be in aridHnetical progreffiolt, the denfities will b6 \%
geometrical progreffion. And fo in infinittmu Thefi; things
twfl bfe fo, when thte denfity of the fluid condenfed by ccfmi^
)>reffi6n is las the foi^e of oompreffion ; or, which is the imbb-
ibing, when the Q>ace pdffefled by the IRofd is lec^ocaily sji
this force. XMtet laws of condenfation may be fuj^fed, ato
that the cube of the compreffing forc^ may be as the iiiqiia^-
^.
Go MATHEMATICAL PHINCIPLES Book II.
drate of thedeniity ; or the triplicate ratio of the force the
fame wilh the qmidruplicate ratio of the deufity: Id which
cafe, if the gravity be reciprocally as the i'quare of the diftance
from the centre, the denfity will be reciprocally as the cube
ot the diliance. Siippofe that the cube of the comprefling^
force be as the quadrato-cube of the dciifity ; and if the gra-
vity be reciprocally as the fquarc of the diftance, thedeulity
will be reciprocally in a fcf(iui plicate ratio of the diftance.
Soppofe the comprcffing force to be in a duplicate ratio of the
denfity, and the gravity reciprocally in a duplicate ratio of the
diftance, and the denfity will be reciprocally as the diftance.
To run over all the cafes that might be offered would be te-
dious. But as to our own air, this is certain from experiment,
that its denfity is either accurately, or very nearly at leaft, as
the compreffing force ; and therefore the denfity of the air in
the atmofphere of the earth is as the weight of the whole in-
cumbent air, that is, as the height of the mercury id theiba-
rometer.
PROPOSITION XXIII. THEOREM XVm. ''
If a fluid be compofed of particles mutually Jiying each- other,
and the denfity be as the camprejjlion, the centrifugal forces
of the particles will be reciprocally proportional to the dijir
ances of their centres. And, vice verfa, particlesjlying each
other with forces that are reciprocally proportional to the
dijiances of their centres, compofe an elajlicjtuid, whofe den-^
jity is as the comprejfion. (PI. 5, Fig. 4.)
- Let the fluid be fuppofed to be included in a cubic fpace
ACE, and then to be reduced by compreffion into a lefteir
cubic i\)ace ace ; and the diftances of the particles retainiqg
a like fituation with refpedl to each other in both the fpace9#
•will be as the fide» AQ, ab of the cubes ; and the denfitiesf of
4he mediums will be reciprocally as the containing fpaq<^
AW, ab' . In the plane fide of the greater cube ABCD take
the fquare DP equal to the plane fide.db of the leffer cube^
.^nd, by the fuppofition, the preffure with which the fquare
DP urges the inclofed fluid will be to the preffure witji w^idi
Ahat fquare db urges the inclofed fluid asthedenfities.of .the
mediums are to each other, that is, as abf.to AB^ J^yt the
SeS. V. OF NATURAL PHILOSOPHY. 6l
pfeffure with which the fquare DB urges the included fluid is
to the preflbre with which the fquare DP urges the fame fluid
as the fquare DB to the fquare DP, that is, as AB* to ab*.
Thereforie, ex aquo, the prefluue with which the fquare DB
urges the fluid is to the preflure with which ihe fquare db
urges the fluid as ab to AB. Let the planes FGH, fgh, be
drawn through the middles of the two cubes, and divide the
fluid into two parts. Thefe parts will prefs each other mu-
tually with the fame forces with which they are themfelves
prefled by the planes AC, ac, that is, in the proportion of ab
to AB : and therefore the centrifugal forces by which thefe
preflures are fufiained are in the fame ratio. The number
of the particles being equal, and the fituation alike, in both
cubes, the forces which all the particles exert, according to
the planes FGH, fgh, upon all, are as the forces which each
exerts on each. Therefore the forces which each exerts on
each, according to the plane FGH in the greater cube, are to
the forces which each exerts on each, according to the plane
fgh in the lefler cube, as ab to AB, that is, reciprocally as
the difliances of the particles from each other. Q.E.D.
And^ vice verfa, if the forces of the Angle particles are re-
ciprocally as the diftances, that is, reciprocally as the iides
of the cubes AB, ab ; the fums of the forces will be in the
fame ratio, and the preflures of the fides DB, db as the fums
of the forces ; and the preflure of the fquare DP to the pref-
fure of the fide DB as ab* to AB*. And, ex aquo, the pref-
fure of the fquare DP to the preflure of the fide db as ab^ to
AB' ; that is, the force of cprnpreflion in the one to the force
of compreflion in the other as the denfity in the former to
the denfity in the latter. Q.E.D.
SCHOLIUM.
By a like reafoning, if the centrifugal forces of the parti-
cles are reciprocally in the duplicate ratio of the difl;ances be-
tween the centres, the cubes of the compreflTmg forces will be
as the biquadrates of the denfities. If the centrifugal forces
be reciprocally in the triplicate or quadruplicate ratio of the
diftances, the cubes- of the comprefling forces will be as the
quadrato-Gubes, oi: ccdbo-cubes of the denfities. And univer-
62 MATHEMATICAL PIINCIPLKS ' Book II.
fally/ if D be put for the diftaoce^ and E for the denfitj of
the compreiTed fluids and the centrifugal forces be reciprocaRUj
as any power D<* of the diftance, whofe index is tbe numfacar
n^ the comprefling forces will be as the cube roots of the
power £*+% whofe index is the number n + 2: and the ooii-'
traiy. AH thefe things are to be underftood of particles whofe
centrifiigal forces terminate in thofe particles that are next
them^ or are dif&ifed not mach further. We have an example
of this in magnetical bodies. Their attra6Uve virtue is termir
nated nearly in bodies of their own kind that are next iheai.
The virtue of the magnet is contradied by the interpofilion of
an iron plate^ and is almoft terminated at it ; for bodies fiiD-
theroff are not attradled by the magnet fo madias by the
iron plate. If in this manner particles repd others of their
own kind that lie next them^ but do not exert their virtue oa
the more remote^ particles of this kind will compofe fiich
fluids as are treated of in this propofiUon. If the virtoe of
any particle diffufe itfelf every way in in^mium, there will he
required a greater force to produce an equal coodeniSatiott of
a greater quantity of tbe fluid. But whether elaftic fluids do
really confift of particles fo repelling each other, is afdijfical
queftion. We have here demouftrated mathematically the
property of fluids coniifting of particles of this kind, that
hence pbilofophers may take occafiou to difcufs that ^iiffC-
iion.
sEcrrioN vi.
Of the motion and refijlance offimependulom bodi&u
PROPOSITION XXIV. THEOREM XIX.
The quantities of matter in funependulous bodies, whofe centrti
of ofcillation are equally dijlantfrom the centre ofjufiftrnjiimf
are in a ratio compounded of the ratio of the weights and the
dtqfHcate ratio of the times of the ofciUatiom in vacuo.
For the velocity which a given force can generate in %
given matter in a given time is as the force and die time di*
re<^y^ and the matter inverfely. The greater the force or the
time is^ or the lefs the matter^ th&greater vdocity will be ge-
nerated. This is manifejft from the fecond law of motion.
Now if pendulums are of the fieone lengthy the motiive loroes
Se&. VI. OF NATURAL PHILOSOPHY. 63
in places equally diftant from the perpendicular are as the
weights : and therefore if two bodies by ofcillating defcribe
equal arcs^ and thofe arcs are divided into equal parts ; fince
.the times in which the bodies defcribe each of the corre-
fpondent parts of the arcs are as the times of the whole ofcil-
lations^ the velocities in the correfpondent parts of the ofciU
lations will be to each other as the motive forces and the
whole times oPtbe ofcillations dire6iJy^ and the quantities of
matter reciprocally : and therefore the quantities of matter are
as the forces and the times of the ofcillations diredily and the
velocities reciprocally. But the velocities reciprocally are as
the times^ and therefore the times diredUy and the velocities
reciprocally are as the fquares of the times ; and therefore
the quantities of matter are as the motive forces and the
fquares of the times^ that is^ as the weights and the fquares of
the times. Q.E.D.
Cor. ]. Therefore if the times are equals the quantities of
matter in each of the bodies are as the weights.
Cob. 2. If the weights are equals the quantities of matter
will be asjthe fquares of the times.
CoR. d. If the quantities of matter are equal, the weights
will be reciprocally as the fquares of the times.
CoR. 4. Whence finoe the fquares of the times, ceteris
paribus, are as the lengths of the pendulums, therefcn^ if both
the times and quantities of matter are equal, the weight! will
'''be as the lengths of the pendulums.
CoR. 5. And univerfally, the quantity of matter in the
pendulous body is. as the weight and the fquare' of the time
dire^ly, and the length of the pendulum inverfely .
Cob. 6. ^tin a non^refifting medium, tlie quantity of
matter in the pendulous body h as the comparative weight
and the fquare of the time diredlly, and the length of the pen-
dttbioi inverfely. For the comparative weight is the motive
force of the body in any heavy medium, as was fliewn above ;
and therefore does the fame thing in fuch a non-refifting me-
dioBi as the ablbhite weight does in a vacuum.
CoR. 7. And hence appears a method both of comparing
liodies 0Q« amoi^ mother^ as to the > quantity of matter iu
64 MATHEMATICAL |>R1NC1PLES Book II,
each ; and of comparing the weights of the fame body in
ttifferent places, to know the variation of its gravity. And
by experiments made with the greatcft accuracy, I have al-
ways found the quantity of matter in bodies to be proportional
to their weight.
PROPOSITION XXV. THEOREM XX.
FunepcnduloHS bodies that are, in atiy medium^ refijied in the
ratio of the moments of time, and futiependulous bodies that
move in a non-rejijling medium of the fame fpecijic gravity,
perform their of dilations in a cycloid in the fame time, and
defcribe proportional parts of arcs together. (PI. 5, Fig. 5.)
Let AB be an arc of a cycloid, which a body D, by vi-
brating in a non-reiifting medium, (hall defcribe in any time.
Bifedl that arc in C, fo that C may be the loweft point
thereof; and the accelerative force with which the body h
urged in any place D, or d, or E, will be as the length of the
arc CD, or Cd, or CE. Let that force be expreffed by that
fame arc ; and fince the refiflance is as the moment of the
time, and therefore given, let it be expreffed by the given
part CO of the cycloidal arc, and take the arc Od in the
fame ratio to the arc CD that the arc OB has to the arc CB:
and the force with which the body in d is urged in a refifting
medium, being the excefs of the force Cd above the refiftance
CO, will be expreffed by the arc Od, and will therefore be
to tlife force with which the body D is urged in a non-refifting
medium in the place D, as the arc Od to the arc CD ; and
therefore alfo in the place B, as the arc OB to the arc CB.
Therefore if two bodies D, d go from the place B, and are
urged by thefe forces ; fince the forces at the beginning arc
as the arcs CB and OB, the firfl velocities and arcs firft de-
fcribed will be in the fame ratio. Let thofe arcs be BD and
Bd, and the remaining arcs CD, Od, will be in the fame ra-
tio. Therefore the forces, being proportional to thofe arcs
CD, Od, will remain in the fame ratio as at the beginnings
and therefore the bodies will continue defcribing together arcs
in the fame ratio. Therefore the forces and velocities .And
the remaining arcs CD, Od, will be always as the whole arcs
CB, QP, and therefore thofe remaining arcs wiU be defcribed
Seil. VI. OF NATURAL PHILOSOPHY. 65
together. Therefore the two bodies D and d will arrive to-
^gether at the places C and O ; that which moves in the non«
refifting medium^ at the place C^ and the other^ in the re-
jifting medium^ at the place O. Now fince the velocities in
C and O are ^s the arcs CB, 01^ the aks which the bodies
defcribe when they go farther will be in the fame ratio. Let
tbofearcsbe CEandOe. The force with which the body
J),m,a,non-refi{ling medium is retarded in E is as- CE, and .
jllie.force with which the body d in the refifting medium
is Fetdrded in e, is as the fum of the force Ce and the refift-
. a^ce CO, that is, as Oe ; and therefore the forces with which
tbe bodies are retarded ai;e as the arcs CB, OB, proportional
tothciarcs CE,'Oe; and therefore the velocities, retarded in
that given ratio, . retmain in the fame given ratio. Therefore
-the velocities and the arcs defcribed with thofe velocities are
.4^ ways to each other in that given ratio of the alts CB and
, OB;; and therefore if the entire arcs AB, aB are taken in the
fwie. ra^tio, the bodies D and d will defcribe thofe arcs toge-
ther, -pmd jn the places A and a will lofe all their motion to-
gether. Therefore . the whole ofcillations are ifochronal,. or
i,are perform^, in equal times; and any parts of the arcs^ as
,lBDf Bd, or BE,: ?ie, that aredefcribed together, arepropor-i /"
tiotial to the whole arcs BA, Ba. Q.E.D.
CoR. Therdpre the fwifteft niotion in a reiifling medium
does not fall upon the'lowefi; ppint C, but is found 9kjthat
point 0> in which the whole arc defcribed Ba is billed.
And .the body proceeding from thence to a, is rietarded at the
fame rate with^iirhich it was accelerated before in its defcent
from B to O.
PROPOSITION XXVI. THEOREM XXI.
Funependulotis liodies, that are rejijiedin the ratio of the velo^
dty, have their pfcillatipns in a cycloid ifochronal.
Tot if two bodies, equally diftant from their centres of fuf-
penfion,. defcribe, in ofcillating, unequal arcs, and the velo-
cities in the correfpondent parts of. the arcs be to each other
as the whole arcs;' the refiftances,. proportional to the velo-
cities. wiU be alfo to each other as the fame arcs. Therefore
if tbefe refiftances be fubdudled from or added to the motive
Vol. II, F
6(J MATHEMATICAL PRIl^ClPLES 6ook 11.
forces arifmg from gravity which are as the fame arcs^ the
differences or fums will be to each other in the fame ratio of
the arcs ; and fince the increments and decrements of the ve-
locities are as thefe differences or fums^ the velocities wUl be
always as the whole arcs ; therefore if the velocities are in aify
one cafe as the whole arcs^ they will remain always in the
fame ratio. But at the beginning of the motion^ when the
bodies begin to defcend and defcribe thofe arcs^ the forooBy
which at that time are proportional to the arcs^ will generate
velocities proportional to the arcs. Therefore the velocities
will be always as the whole arcs to be defcribed^ and thei^
fore thofe arcs will be defcribed in the fame time. Q.E.D,
PROPOSITION XXVII. THEOREM XXIL;
If funependulous bodies are rejijied in the duplicate ratio rf
their velocities, the differences between the times of the ofctt"
lations in a refijling medium, and the times of the ofciliaiiom
in a non-rejijling medium of the fame fpecific gravity, wsll
be proportional to the arcs defcribed in ofcillating nearly,
" For let equal pendulums in a refifting medium defcribe the
unequal arcs A^ B ; and the reiiflance of the body in the arc
'A will be to the refiftance of the body in the correfpondeat
part of the arc B in the duplicate ratio of the velocities^ that
is, as A A to BB nearly. If the refiflance in the arc B were
to the refiftance in the arc A as AB to AA, the times in the
ar^^ A and B would be equal (by the laft prop.) Therefore
the refiftance AA in the arc A, or AB in the arc B, caufes the
excefs of the time in the arc A above the time in a non-refifi-
ing medium; and the refiftance BB caufes the excefs of the
time in the arc B above the time in a non-refifting m^iiim.
But thofe exceffes are as the efficient forces AB and BB
uedrly, that is^ as the arcs A and B. Q.E.D.
CoR. I. Hence from the times of the ofcillaUons in un-
equal arcs in a refifting medium, may be known the times of
the ofcillations in anon-refifling medium of the fame fpecific
gravity. For the difference of the times will be to the excefii
of the time in the leffer arc above the time in a non-refifUog
Medium as the difference of the arcs to the leffer arc.
Se£t. VI. OF NATURAL PHILOSOPHY. 67
Cor. 2. The (horter ofcillations are more ifochronal^ and
▼ery ihort ones are performed nearly in the fame times as in a
non-refifiipg medium. But the times of thofe which are per-
formed in greater arcs are a little greaier^becaufe the reiift-
ance in the defcent of the body^ by whiSi the time is pro-
longed^ is greater^ in proportion to the length defcribed in the
defcent than the refiftance in the fubfequent afcent^ by
which the time is contradied. But the time of the ofcilla-
tions^ both (hort and long^ feems to be prolonged in fome mea-
fure by the motion ^of the medium. For retarded bodies are
refifted fomewhat lefs in proportion to the velocity^ and ac-
celerated bodies fomewhat more than thofe that proceed uni-
fojrmly forwards; becaufe 4he medium^ by the motion it has
received ^m the bodies^ going forwards the fame way
with them^ is more agitated in the former cafe^ and lefs in the
latter ; and fo confpires more or lefs with the bodies moved.
Therefore it refifls the pendulums in their defcent more^ and
in their afcent «lefs^ than in proportion to the velocity; and
thefe two caufes concurring prolong the time.
PROPOSITION XXVIII. THEOREM XXIII.
If a funependuiom body, of dilating in a cycloid, be refilled
in the ratio of the momeni9 of the time, its refiftance zdil be
to the force of gravity as the txcefs of tli^ arc defcribed in
the whole defcent above the arc defcribed in the fubfequent
afcaU to twice the length of the pendulum. (PI. 5, Fig^^O
Let BC reprefent the arc defcribed in the defcent, Ca the
arc defcribed in the afcent> and Aa the difference of the arcs:
and things remaining as they were conftruAed and demon-
fbrated in prop. 25^ the force with which the ofcillating bod/
is urged in any place D will be to the force of refiftance as
the arc CD to the arc CO^ which is half of that difference
Aa. Therefore the force with which the ofcillating body is
uiged at the beginning or the higheft point of the cycloid^
that is, the force of gravity, will be to the refiftance as the
arc of the cycloid, between that higheft point and loweft
point C, is to the arc CO; that is (doubling thofe arcs), as the
whole cycloidal arc, or twice the length of the pendulum, to
^e arc Aa. Q.E.D.
F 2
I
■
68 MATHEMATICAL PIllNCIPLES JBooJIt II.
PROPOSmON XXIX. PROBLEM VT.
Suppojing that a bodf/ ofcillating in a cjfcloid is refined hi
duplicate ratio of the velocity : to find the rrfijtance in- raeA
place. (PI. 5, Eg. 6.)
Let Ba be an arcdefcribcd in one entire ofcillation, C thii
lowed point of the cycloid, and CZ halt' the whole cycloidftl
arc^ equal to the length of the pendulum ; and let it be re- .
quired to find the redftance of the body in any place D. Cat
the indefinite right line OQ in the points O^ S, P, Q, to that
(ereding the perpendiculars OK, ST, PI, QE, and with the
centre O, and the afymptotes OK,OQ, defcribing the hyper**
bolaTIGE cutting the perpendiculars ST, PI, QE in T, I, apd
E, and through the point I drawing KF, parallel to the alymp-
tote OQ, meeting the afymptote OK in K, and the perpendi*
culars ST and QE in Land F) the hyperbolic area PIEQ may
be to the hyperbolic area PITS as the arc BC, defcribed in the
defcent of the body, to the arc Ca defcribed in the afoent;
and that the area lEF may be to the area ILT as OQ»to OS:
Then with the perpendicular IMN cut off the hyperbolic «re8
PINM, and let that area be to the hyperbolic area PIEQ as
the arc CZ to the arc BC defcribed in the defcent. And if
the perpendicular RG cut off the hyperbolic area PIGR,
which fhall be to the area PIEQ as any arc CD to the arc
BC defcribed in the whole defcent, the refiftanceia
any- place D will be to the force of gtavity as the area
^ lEF — IGH to the area PINM.
For fince the forces arifing from gravity with which tlie
body is urged in the places Z, B, D, a, are as the arcs CZ, ^
CB, CD, Ca, and thofe arcs are as the areas PINM, PIEQ,
PIGR, PITS; let thofe areas be the exponents both of the
arcs and of the forces refpeAively. Let Dd be a very fmall
fpace defcribed by the body in its defcent ; and let it be ez-
preffed by the very fmall area RGgr comprehended between
the parallels RG, rg ; and produce rg to h, fo that 6Hh]g;
and RGgr itiay be the contemporaneous decrements of th«
Rr
areas IGH, PIGR. And the increment GHhg— ^q lEF,
I
ScS. VI. OP NATUBAL PHILOSOPHY. 69
or Rr X HG -^ g^ lEF, of the area gg lEF — IGH will
be to the decrement RGgr, or Rr x RG, of the area PIGR,
" lEF
as HG — ^Q- to RP; and therefore^ OR x HG —
OR
^g lEP to OR X GR or OP x PI, that is (becaufe of the
equal quantities OR x HG, OR x HR — OR x GR,.
ORHK ~ OPIK, PIHR and PIGR + IGH), as PIGR + .
OR OR
IGH — -^ lEF to OPIK. Therefore if the area ~ lEF
— , IGH be called Y, and RGgr the decrement of the area
PIGR be given, the increment of the area Y will be as PIGR
_y.
Then if V reprefent the force arifing from the gravity, pro-
portional to the arc CD to be defcribed, by which the body is
ai9jed upon in D, and R be put for the reiiftance, V — R will
be the whole force with which the body is urged in D.
Therefore the increment of the velocity is as V — R and the
particle of time in which it is generated conjundly. But the
velocity itfelf is as the contemporaneous increment of the
fpace defcribed dire<5Uy and the fame particle of time inverfe-
^y. Therefore, fince the refiftance is, by the fuppoiition, as
the fquare of the velocity, the increment of the refiftance will
(by lem. £) be as the velocity and the increment of the v^k>ci-
ty conjundlly, that is, as the moment of the fpace and V — R
conjunctly; and, therefore, if the moment of the fpace be
given, as V — R; that is, if for the force V we put its expo*
nent PIGR, and the refiftance R be expreffed by any other
area Z, as PIGR —Z. ,
Therefore the area. PIGR uniformly decreafing by the fub-^
du^lion of given moments, the area Y increafes in proportion
of PIGR— Y, and ihe area Z in proportion of PIGR — Z.
And therefore if the areas Y and Z begin together, and at
the beginning are equal, thefe, by the addition of equal mo*
ments, will continue to be equal; and in like manner decreaf-
ing by equal moments, will vaniCh together. And, vice verfa,
if they together begin and vanifti, they will have equal mo-
F 3
70 MATHEMATICAL PBINCIPLES Book IL
ments and be always equal; and tliat^ becaufe if the rciiftanoe
Z be augmented, the velocity together with the arc Ca, defcrib-
ed in the afcent of the body, will be diminifhed ; and the point
in. which all the motion together with the rcfiftance cea(ei
coming nearer to the point C, the refinance vanifhes fooner
than the area Y. And the contrary will happen when the
refiftance is diminiflied.
No w the area Z begi ns and ends where the rellftance is nothing,
that is, at the beginning of the motion where the arc CDis eqnal
to the arc CB, and the right line RG falls upon the right line QE;
and at the end of the motion where the arc CD is equal to die arc
OR
Ca,and RG falls upon the right line ST. And the area Yor ^r
lEF — IGH begins and ends alfo where the refiftance b no-
thing, and therefore where j^ lEF and IGH are equal;
that is (by the conftrudlion), where the right line RG 'falb
fucceffively upon the right lines QE and ST. Therefore
thofe areas begin and vanifh together, and are therefore al-
OR
ways equal. Therefore the area tt^ IEF — IGH is equal
to the area Z, by which the refiftance is exprefled, and there-
fore is to the area PINM, by which the gravity is exprefled,
as the refiftance to the gravity. Q.E.D.
Cor. 1. Therefore the refifl:ance in the loweft place C it
OP
to the force of gravity as the area tt^ IEF to the area
PINM.
CoR. 2. But it becomes greateft where the area PIHR b
to the area IEF as OR to OQ. For in that cafe its mo-
ment (that is, PIGR — Y) becomes nothing.
Cor. 3. Hence alfo may be known the velocity in each
place, as being in the fubduplicate ratio of the refiftance, and
at the beginning of the motion equal to the velocity of the
body ofcillating in the fame cycloid without any refift^nbe.
However, by reafon of the difficulty of the calculation by
which the refifianoe and the velocity are found by this pftH
-Tfalc/ . i f>f. If. nrMii'.vi:.
\F^^.
ty%
Z Y X
U^»-3,
I ■
\^
M
*V
^
^.
?S
^
'-. £
= J
(.f> }
T
My
tl.
H
\
J'
■ r
*"
K
> M
&5. VI. OP NATURAL PHILOSOPHY. 71
poiitioD, we have thought fit to fubjoin the propofition
foUovring.
PROPOSITION XXX. THEOREM XXIV.
If a right line aB (PI. 6, Fig. 1) be equal to the arc of a cy-
cloid which an ofcillating body defcribe^, and at each of its
points D the perpendiculars DK be erected, which Jhall be to
the length of the pendulum as tfte refijiance of the body in
the correfponding points of the arc to the force of gravity;
I fay, that the difference between the arc defcribed in the
whole defcent and the arc defcribed in the whole fubfequent
afcent drawn into half the fum of the fame arcs will be equal
to the area BKa which all thofe perpendiculars take up.
Let the. arc of the cycloid^ defcribed in one entire ofcillation^
be exprelTed by the right line aB^ equal to it^ and the arc
which would have been defcribed in vacuo by the length AB.
BiCe(5l AB in C, and the point C will reprefent the loweft
point of the cycloid^ and CD will be as the force arifing from
gravity^ with which the body in D is urged in the diredlion of
the tangent of the cycloid^ and will have the fame ratio to
the length of the pendulum as the force in D has to the force
of gravity. Let that force, therefore, be exprefTed by that
length CD, and the force of gravity by the length of the pen-
dulum ; and if in D£ you take DK in the fame ratio to the
length of the pendulum as the refiftance has to the gravity,
DK will be the exponent of the refiftance. From the centre
C with the interval CA or CB defcribe a femi-circle BEeA.
Let the body defcribe, in the leaft time, the fpace Dd ; and,
erecting the perpendiculars DE, de, meeting the circumfe-
rence in E and e, they will be as the velocities which the bo-
dy defcending in vacuo from the point B would acquire in the
places D and d. This appears by prop. 52, book I. Jjet,
therefore, thefe velocities be exprefTed by thpfe perpendic^lai*s
DE, de ; and let DF be the velocity which it acquires in D
by falling from B in the refifting medium. And if frpm the
centre C with the interval CF lye defcribe the circle FfM
meeting the right lines de and AB in f and M, then M will
be the place to which it would thenceforward, without far-
ther refiftance^ afcend^ and df the velocity it would acquire in
F4
I
i
«
72 MATHRMATTCAL PRINCIPLES Bo6k \[.
d. . Whence, alfo, if Fg reprelcnt Uie moment of the velocity
which the body D, in dcfcribing the leail fpacc Dd, lofes by
the refiftancc of the medium; and CN be taken equal to Cg;
then will N be the place to which the body, if it met no far-
ther refiftance, would thenceforward afcend, and MN will be
the decrement of the afccnt arifingfrom the lofs of that velo-
city. Draw Fm perpendicular to df, and the decrement Fg of
the velocity DF generated by the refiftance DK will be td the
increment fm of the fame velocity, generated by the forceCt),
as the generating force DK to the generating force CD. But
becaufe of the fimilar triangles Fmi^ Fhg, FDC, fm is to Fm
or Dd as CD to DF; and, ex aquo, Fg to Dd as DK to DF.
Alfo Fh is to Fg as DF toCF; and> ex aquo periurbati,¥h: or
MN to Dd as DK to CF or CM; and therefore the fum of aJl
the MN X CAI will be equal to the fum of all the Dd X DK^
At the moveable point M fuppofe always a re6hingalar — "'
"nate creeled equal to the indeterminate CM, which by a con-
tinual inotion is drawn into the whole length Aa; and the tra*
pezium defcribed by that motion, or its equal, the rectangle Aa
X f aB, will be equal to the fum of all the MN X CM, and
therefore to the fum of all the Dd x DK, that is, to the'area
BKVTa. Q.E.D*
CoR. Hence from the law of refiftance, and the difterence
Aa of the arcs Ca, CB, may be collefted the proportion of the
refiftance to the gravity nearly.
For if the refiftance DK be uniform, the figure BKTa will
be a re6langle under Ba and DK; and thence the re6langle
under |Ba and Aa will be equal to the redangle under Ba and
DK, and DK will be equal to |Aa. Wherefore fince DK is the
exponent of the refiftance, and the length of the pendulum the
exponent of the gravity, the refiftance will be to the gravity as
|Aato the length of the pendulum; altogether as in prop. 28
is demonftrated.
If the refiftance be as the velocity, the figure BKTa will
be nearly an ellipfis. For if a body, in a nou-refifiing me-^
dium, by one entire ofcillation, (hould defcribe the length
BA, the Telocity in any place D would be as the ordinate
SeSti VI. OF NATURAL PlIlLOSbPHY. ^4
DE of the circle defcribed on the diameter AB. Therefore
iiDce Ba in the reiifting medium^ and BA in the non-refifting
one) .are defcribed nearly in the fame times; and therefore
the velocities in each of the points of Ba are to the veloci-
ties in the correfpondent points'of the length BA nearly as
Ba is tO'BA; the velocity in the point D in the relGfting me-
dium will be as the ordinate of the circle or eliipfis defcribed
upon the diiattieter Ba ; and therefore the figure BKVTa will
be nearly an ellipiis. Since the refiftance is fuppofed propor-
tibnal t6 the velocity, let OV be the exponent of the refift-
ance in the middle point O; and an eliipfis BRVSa defcribed
with the centre O, and the femi-axes OB, OV, will be nearly
e^nal to the figure BKVTa, and to its "equal the re6langle Aa
X BO. Therefore Aa x BO is to OV x BO as the area of
this eliipfis to OV x BO ; that is, Aa is to OV as th^area of
the femi-circle to the fquare of the radius, or as 11 to 7 near-
ly; and, therefore, i^Aa is to the length of the pendulum as
therefifiiance of the ofcillating body in O to its gravity.
Now if the refiftance DK be in the duplicate ratio of the
velocity, the figure BKVTa will be almoft a parabola having
V for its vertex and OV for its axis, and therefore will be
nearly equal to the re6langle under .|-Ba and OV. Therefore
the re6iangle under |Ba and Aa is equal to the reftanglefBa
X OV, and therefore OV is equal to ^Aa ; and therefore the
refiftance in O made to the ofcillating body is to its gravity
as |ula to the length of the pendulum.
And I take thefe conclufions to be accurate enough for
pradlical ufes. For fince an eliipfis or parabola BRVSa falls
in with the figure BKVTa in the middle point V, that figure,,
if greater towards the pari BRV or VSa than the other, is lefs
towards the conftrary part, and is therefore nearly equal to it.
PROPOSITION XXXI. THEOKEM XXV.
If the refiftance made to an ofcillating body in each of the pto^
portional parts of the arcs defcribed be attgmcnted or dimi-
nifhed in a given ratio, the difference between the arc de-
fcribed in the defcent and the arc defcribed in the fubfequent
afcent mil be augmented or dimini/hed in the fame ratio4
74 MATH£MATirAL PAINCIPLES JBooA IL
For thai difference arifes from the retardation of the pen*
duluin b}' the refiflance of the medium, and therefore is as
the whole retardation and the retarding refiftance propor-
tional thereto. In the foregoing proportion the redlangle
under the right line {aB and the difference Aa of the area
CB, Ca, was equal to the area BKTa. And that area^ if the
length aB remains^ is augmented or diminiflied in the ratio
of the ordinates DK ; that is^ in the ratio of the refiftancCj
and is therefore as the length aB and the refiftance con-
jandtly. And therefore the rei^angle under Aa and {aB is aft
aB and the refiftance conjundly^ and therefore Aa is as the
refiftance. Q.E.D.
Cor. 1. Hence if the refiftance be as the velocity^ the di&>
ference of the arcs in the fame medium will be as the whol^
arc defcribcd: and the contrary.
CoK. 2. If the refiftance be in th6 duplicate ratio of the
velocity^ that difference will be in the duplicate ratio of the
whole arc: and the contrary.
CoR. 3. And univerfally^ if the refiftance be in the tripli-
cate or any other ratio of the velocity, the difference will be
in the fame ratio of the whole arc: and thje contrary.
Cor. 4« If the reftftance be partly iu the ftmple ratio of
the velocity^ and partly in the duplicate ratio of the fame^
the difference will be partly in the ratio of the whole arc^
and partly in the duplicate ratio of it: and the contrary.
So that the law and ratio of the refiftance will be the
fame for the velocity as the law and ratio of that difference
for the length of the are.
CoR. 6. And therefore if a pendulum defcribe fucceffively '
unequal arcs^ and we can find the ratio of the increment or
decrement of this difference for the length of the arc defcrib-
edj there will be had alfo the ratio of the inqreinent or de*
crement of the refiftance for a greater or lefs velocity.
GENERAL SCHOLIUM.
From thefe propofitions we may find the refiftance of me-
diums by pendulums ofcillating therein. I found the refift-
ance of the air by the following experiments. I fufpended a
wooden globe or ball weighing .57^V ounces troy, its diame*
tcr 6r djondon inches, by a fine thread on a firm hook, fo that
Se&.Vl. OF NATURAL PHILOSOPHY. 76
the diifamce between the hook and the centre of ofcillation of
the globe was 10| feet. I marked on the thread a point lO
feet and 1 inch diftantfrom the centre of fufpeniion; and
even with that point I placed a ruler divided into inches^ bj
the,herp whereof I obferved the lengths of the arcs defcribed
by the pendulum. Then I numbered the ofcillations in
which the globe would. lofe j- part of its motion. If the pea- •
dulum was drawn afide from the perpendicular to the dif-
tdnce of £ inches^ and thence let go^ fo that in its whole de-
fcent it defcribed an arc of 2 inches^ and in the firft whole of-
cillation^ compounded of the defcent and fubfequent afcent^
an arc of almoft 4 inches^ the fame in 164 ofcillations' loft
jr part of its motion^ fo as in its laft afcent to defcribe an arc
of li inches. If in the firft defcent it defcribed an arc of 4
inches^ it loft {• part of its motion in 121 ofcillations^ fo as in
its laft afcent to defcribe an arc of 3| inches. If in the firft
defcent it defcribed an arc of 8, l6, 32, or 64 inches, it loft {.
part of its motion in 69, 35|, lB|,9y ofcillations, refpedively.
Therefore the difference between the arcs defcribed in the firft
defcent and the laft afcent was in the 1ft, 2d, 3d, 4th, 5th,
6th cafes, ^, |, J, 2, 4, 8 inches, refpedlively. Divide thofe
differences by the number of ofcillations in each cafe, and in
one mean ofcillation, wherein an arc of 3|,7|, 15, 30, 60, 120
inches was defcribed, the difference of the arcs defcribed in
the defcent and fubfequent afcent will be -g-f^, -y^ -jV' "siV*
-sV^ i'l' parts of an inch, refpe6iively. But thefe differences
in the greater ofcillations are in the duplicate ratio of the arcs
defcribed nearly, but in leffer ofcillations fomething greater
than in that ratio ; and therefore (by cor. 2, prop. 31 of this
book) the refiftance of the globe, when it moves very fwift, is
in the duplicate ratio of the velocity, nearly; and when it
moves flowly, fomewhat greater than in that ratio.
Now let V reprefent the greateft velocity in any ofcillation,
and let A, B, and C be given quantities, and let us fuppofe ihc
difference of the arcs to be AV + BV^ + CV*. Since the
greateft velocities are in the cycloid as 1 the arcs defcribed in
circillating; and in the circle as | the chords of th6fe arcs; and
76 MATHEMATICAL PRINCIPLES Book IL
therefore in equal arcs are greater in the c}xloid than in the
circle in the ratio of ^ the arcs to their chords; hut the times
in the circle are greater than in the cycloid, in a reciprocal
ratio of the velocity ; it is plain that the differences of the arcs
(which are as the refiftance and the fquare of the time con-
jun<91y) are nearly the fame in both curves: for in the cy»
cloid thofe differences mud be on the one hand augmentcxi,
with the refiftance^ in about the duplicate ratio of the arc to
the chords becaufe of the velocity augmented in the fimple
ratio of the fame; and on the other hand diminiihed, with
' the fquare of the time^ in the fame duplicate ratio. There-
fore to reduce thefe obfervations to the cycloid^ we muft take
the fame differences of the arcs as were obferved in the cir-
cle^ and fuppofe the greatcft velocities analogous to the half,
or the whole arcs^ that is^ to the nun^bers i, I, 2, 4^ 8, 16.
Therefore in the 2d^ 4th^ and 6th cafes^ put 1^ 4, and l6 for
V; and the difference of the arcs in the 2d cai'e will become
— = A + B + C; inthe 4th cafe, -zrr^^'^A + 8B +
16C; in thie Ci^h cafe, ^ = l6A + 64B + 256C. Thefe
equations reduced give A = 0,0000916, B = 0,0010847,
and C zz OftOQ955S. Therefore die difference of the arcs is
as 0,0000916V + 0,001 0847 V^ + 0,00*29558 V» : and there-
fore fince (by cor. prop. 30, applied to this cafe) the refift-
ance of the globe in the middle of the arc defcribed in ofcil-
lating, where the velocity is V, is to its weiglit as -^AV +
^BV^ + |CV* to the length of the pendulum, if for A, B^
and C you put the numbers found, the refiftance of the globe
will be to iU weight as 0,0000583V + 0,0007593V^ +
0,0022169V* to the length of the pendulum between the cen-
tre of fufpenfion and the ruler, that is, to 121 inches. There-
fore fince V in the 2d cafe reprefents 1, in the 4th cafe 4, and
in the 6th cafe r6, the refiftance will be to the weight of the
globe, in the 2d cafe, as 0,0030345 to 121; in the 4th, as
0,041748, to 121; in the 6th, as 0,61705 to 121.
S€6l. VI. OF NATURAL PHILOSOPHY. 77
The arc which the point marked .in the thread defcribed
8
in ihe 6th cafe^ was of 120 — --t, or 1 19^ inches. And
therefore fince the radius was 121 inches^ and the length of
the pendulum between the point of fufpenfion and the centre
of the globe was 126 inches, the arc which the centre of the
globe defcribed was 124"A inches. Becaufe the greateft velo*
city of the bfcillating body, by reafon of the refiftance of the
air, does not fall dn the lowed point of the arc defcribed, but
near the middle place of the whole arc, this velocity will be
nearly the fame as if the globe in its whde defcent in a non-
refifiing medium fhould defcribe 6^^ inches, the half of that
arc, and that in a cycloid, to which we have above reduced
the motion of the pendulum ; and therefore that velocity will
be equal to that which the globe would acquire by falling per-
- pendiculaiiy from a height equal to the verfed fine of that arc.
But that' verfed fine in the cycloid is to that arc 62-g^ as the
lame arq to twice the length of the pendulum 252, and there-
fore equal to 15,278 inches. Therefore the velocity of the
pendulum is the fame 'Which a body would acquire by falling,
and in its fall defcribing a fpace of 15,278 inches. Therefore
with fuch a velocity the globe meets with a refiftance which
is to its weight as 0,61705 to 121, or (if we take that part only
of the refiftance which is in the duplicate ratio of the velocity)
as 0,56752 to 121.
I found, by an hydroftatical experiment, that the weight of
this wooden globe was to the weight of a globe of water of the
fame magnitude as 55 to 97 : and therefore fince 121 is to
213>4 in the fame ratio, the refiftance made to this globe of
water, moving forwards with the above-mentioned velocity, will
be to its weight as 0,56752 to 213,4, that is, as 1 to 376^.
Whence fince the weight of a globe of water, in the time in
which the globe with a velocity unifornJy continued defcribes
a length of 30,556 inches, will generate all that velocity in the
falling globe, it is manifeft that the force of refiftance uni-
formly continued in the fame time will take away a velocity,
which will be tefir than the other in the ratio of 1 to 376xVf
78 MATHEMATICAL PRINCIPLES Book IL
that is, tlie Tr^j; part of tlie whole velocity. And therefore
in the time that the globe, witli the fame velocity uniformly
continued, would defcribe the length of its femi-diameter, of
S^r inches, it would lofe the 77^ part of its motion.
I alfo counted the ofcillations in which the pendulum loft
^ part of its motion. In the following table the upper num-
bers denote the length of the arc defcribed in the firfl defoentj
exprefled in inches aqd parts of an inch ; the middle numbers
denote the length of the arc defcribed in the laft afcent ; and
in the lowed place are the numbers of the ofcillations. I ^ve
an account of this experiment, as being more accurate than
that in which only j- part of tlie motion was loft* I leave the
calculation to fuch as are difpofed to make it.
Firjidefcent 2 4 8 16 32 64
Laft afcent 1|. 3 6 12 fi4 48
Numb.ofofcill... 374 272 162). SSi 4\^ 22f
I afterwards fufpended a leaden globe of 2 inches in diaF*
meter, weighing 9,6^ ounces troy by the famie thread, fo that
between the centre of the globe and the point of fufpenfion
there was an interval of 10} feet, and I counted the ofcil-
lations in which a given p.art of the motion was loft. The firft
of the following tables exhibits the number of ofcillations in
which f part of the whole motion was loft; the fecond
the number of ofcillations in which there was loft ^ part of
the fame.
Firft defcent 1 2 4 8 16 32 64
Laft afcent i J SJ 7 14 28 56
Numb. ofofcilL..Q26 228 193 140 90^ 53 30
Firft defcent ..... 1 ^ 4 ^ 16 32 64
Laft afcent i H 3 6 12 $4 48
Numb, of ofcill... 510 518 420 318 204 121 70
Seledling in the firft table the 3d, 5th, and 7th obfervations^
and expreifing the greateft velocities in thefe obfervations par-
ticularly by the numbers 1, 4, 16 refpeftively, and generally
by the quantity V as above, there will come out in the 3d ob-
fervation ~ =: A 4- B 4- C> in the 5th obfenration rrrr =
1 y o. 9O3*
SeS. VI. OP NATURAL PHILOSOPHY. 79
4A -f- 8B -f 16C, in the 7th obfervation ;^ = ISA + 64B
+ 556C. Thefe equations reduced give A =: 0^001414, B =
0,00p297> C =r O,00O879j And thence tiie refiftance of the
globe moving with the velocity V will be to its weight 26j
ounces in the fame ratio as 0,0009V + 0,000208V^ +
O,O0065gV* to 121 inches, the length of the pendulum. And
if we regard that part only of the refiftance which is in the'
duplicate ratio of the velocity, it will be to the ^i^elght of the
globe as 0,000659V* to 121 inches. But this part of the re-
fiftance in the ift experiment was to the weight of Ihe wooden
globe oi bl4v ounces as 0,0022 17V* to 121 ; and thence the
refiftance ofthe wooden globe is to the refiftance of the leaden!
onp (their velocities being equal) as 57 -i^ into 0,002217 to 26^
.into 0,000659, that is, as 7t to 1. The diameters of the two
globes were 6{- and 2 inches, and the fquares of thefe are to
each other as 47^ and 4, or llf| and 1, nearly. Therefore
the refiftances of thefe equally fwift globes were in lefs than a
duplicate ratio of the diameters. But we have not yet con-
fidered the refift;ance of the thread, which w«is certainly very
confiderable, and ought to be fubdu6led from the refiftance
of the pendulums Here found. I could not determine this ac-
curately, but I found it greater than a third part of the whole
refiftance of the lefler pendulum ; and thence I gathered that
the refiftiances of the globes, when the refiftance of the thread
is fiibdu6fed, are nearly in the duplicate ratio of their dia*
meters. For the ratio of 7^ — ^ to 1 — |, or 10| to 1, is not
very different from the duplicate ratio of the diameters 1 ]f|
to 1.
Since the refifl:ance of the thread is of lefs moment in
greater globes, I tried the experiment alfo with a globe whofe
diameter was 18^ inches. The length of the pendulum be-
tween the point of fufpenfion and the centre of ofcillation
was 122|- inches, and between the point of fufpenfion and the
knot in the thread 109|. inches. The arc defcribed by the
knot at the firft defcent of the penduluu^ was 32 inches. The
arc defcribed by the fame knot in the laft afcent after five of-
80 MATHEMATICAL PK1NCIPLE8 Book 11.
ciUations was 28 inches. The fum of the «ircs^ or the whole
arc defcribed in one mean ofcillation, was 60 inches. The
difference of the arcs 4 inches. The -Ar part of this^ or.Ae
difference between: the defcent and afcent in one mean ofetl-
lation^ is 4 of an inch. Then as the radius 109^ to the radius
I22i, lb is the whole arc of (50 inches defcribed by the knot in
one mean ofciliation to the whole arc of 67i inches defcribed
by the centre of the globe in one mean ofciliation ; and fb is
the difiere.nce ■§■ to a new difference 0^4475. If the length of
thfe arc defcribed were to remain^ and the length of the pen-
dulum (hould be augmented in the ratio of 126 to 122j-> the
time of the ofciliation would be augmented^ and the velocity
of the pendulum would be diminifhed in the fubduplicate of
that ratio ; fo that the difference 0^4475 of the arcs defcribed
in the defcent and fubfequent afcent would remain. Then if
the arc defcribed be augmented in the ratio of 124^ to 67}y
that difference 0^4475 would be augmented in the duplicate of
thatratio^ and fo would become 1^5295. Thefe things would
be fo upon the fuppoiition that the reiiftance of the peudtt-
lu'm were in the duplicate ratio of the velocity. Therefbie if
the pendulum defcribe the whole arc of 124^ inches^ and its
length between the point of fufpenflon and the centre of of«
cillation be 126 inches^ the difference of the arcs defcribed in
the defcent and fubfequent afcent would be 1^5295 inch^.
And this difference multiplied into the weight of the pendu-
lous globe^ which was 208 ounces^ produces 318^136. Again;
in the pendulum above-mentioned^ made of a wooden globe,
when its centre of ofciliation^ being 126 inches from the point
of fufpeniion^ defcribed the whole arc of 124^ inches, the
difference of the arcs defcribed in the defcent and afcent was
126 8
— - in^p — • This multiplied into the weight of the globe,
which was 57-s^ ounces, produces 49,396. But I multiply
thefe differences into the weights of the globes, in order to
find their refiftances. For the. differences arife from the re-
fiftances, and are as the refiftances diredily and the weights
inverfoly. Therefore the refiftances are as the numbers 318,136 '
and 49,396. But that part of the refiftance of the leffer glob^^
<$^$. VI« Ot NATURAL PHIX<OSOPRT< 81;
which is in the duplicate ratio of the velocity, was to the
whole refiftance as 0,56752 to 0,6l675, that is, as 45,453 to
40,996^ whereas that part of the refiftance of the greater
^be is almoft equal to its whole refiftance ; and fo thqfe
parts are nearly as 318,136 and 45,453, that is, as 7 and 1.
But the diameters of the globes are l^^aiid f^; and their
fquaies 351^ and '4711 are as 7^438 and 1, th^t is, as the re-
fiftances of the globes 7 and 1, nearly. The difference of thele
ratios is fcarce greater than may arife from the refiijbance of
the thread. Therefore thofe parts of the refiftances which
ave, when the globes are equal, as the fquares of the velo-
cities, are alfo, when the velocities are equal, as the fquares
of the diameters of the globes. . x. ;
Bot the .gre ateft of the globes I ufed in thefe eicperimepts
was not perfe^y fpherical, and therefore in this calculation
I have, for brevity's fake, negle6led fome little niceties ; being
not very folicitous for an accurate calculus in. an experiment
liiai was apt yeiy accurate. So that I could w|(h .that tji^fe
expeiin^aats ;^^re tried again with other globes, .of. a larger
fi«e,.poreja;iii^imber, and more accurately formed; fince the.
demonfi^tiop.oif a vacuum depends thereon. Jf the globes,
1^ tak^ -fH |k,,geoifietrical proportion, as fuppofe whofe dia-.
meters Afei j(l,:8,. l6, 32 inches-, one may cpllcifl from. the.
{fogreffioB.olifqrved in the experiments what would happen if
the fflobeg were ftill larger.
..In cofder.ta compare the n^fi^ftances of .^iffeijent ^uids witl^j
whotliefp I made the fpUowi^g trials. ](|u:pQ^r€;d ^ wdod^if
veflel.4..feetlong^l foptbroa^, and l.fppt.hig}ii;,,TTiisveffei>
liei^g ,iq;i^veced, I £dled with ipring wat^, ',9,ij^^ (having im*
merf^ p^4ia^m$ th^r^in> Imade them ofq^Uat^i|fij-ithe wfitec^.
And .I)foiiiid tjiat a leaden glob^ weighing. l66j; pun9es,,2^nd
in dianieter 3i inches, moved therein as it is fet, down in the.
ib|lowing.ftabl^ ; the length of the pendulum. from ,t)ie point
of fuipeqfion to a certain point miu:ked in the .tj[»repd<beipg
186 incbfss, and to the cenjtre of ofQill^tion 1;3^ inches.
•f ■!•.- »
I
/
m MATHEMATICAL PKIKCIPLE9 Book TL.
The arc defcribedin
the jirfi defeent by
a point marked in -64 . 32 • 16 • 8 • 4 • fi • 1 • 1 • f
the thread, was
inches
The arc deferibed in'^
the laft afcent am Ud . 24 . 12 . 6 . 3 . It . } • | . ^
inches J
The difference of the \
^m i^ojK^r^ioiw/Lg .8.4.2.1.*.*.*.^
to the motion loftA n ^ ^ -^
was inches J
The number of the'\
ofcillations in wa-V H . l{ . 3 . 7 . IH . 12% . I3i
ter }
The number of the of-^
eiUations in air J ®** • 287 . 535
In the experiments of the 4th column there wett tqmi
motions loft in 535 ofcillations made in the air^ and If in
water. The ofciUations in the air were indeed a little fwifker
than thofe in the water. But if the ofcillations in die wafer
were accelerated in fnch a ratio that the motions of the pen-'
dolums might be equally fwiffc in both mediums^ there would
be fiiil the fame number If of ofcillations in the water^ and
by thefe the fame quantity of motion would be loft as befioie;
becaufe the refiftance is increafed; and the fquare of the thne
diminifhed in the fame duplicate ratio. The pendulums^ there*
lore, being of equal velocities^ there were equal motions loft
in 535 ofcillations in the air^ and If in the water; and them*;
fore the refiftance of the pendulum in the water is to its ie^
fiftance in the air as 535 to Ij-. This is the proportion of Hmn-
whole refiftances in the cafe of the 4th column.
Now let AV + CV^ reprefent the differ^ice of the arcade*
fcribed in the defeent and fubfequent afcent by the globe
moving in air with the greietteft velocity V ; and fince the
greateft velocity is in the cafe of the 4th column to the greateft
velocity in the cafe of the 1ft column as 1 to 8; and that dif-
ference of the arcs in the cafe of the 4th column to the dif-
Sc3. VL OF NATVBAL PHILOSOPHY* 83
2 16
ference in the cafe of the 1ft column as •— - to t— -j or as 85|
to 4280 ; pat in thefe cafes 1 and 8 for the velocities^ and 85^
and 4280 for the differences of the arcs^ and A + C will be
r= 85*, and 8A 4- 64C = 4280 or A + 8C = 535 ; and
then, by reducing thefe equations, there will come out 7C
= 449f and C = 64^ and A =: 2lf ; and therefore the re-
fifiance, which is as t|2^AV + JCV*, will become as IS^V
+ 48^V\ Therefore in the cafe of the 4th column, where
the vdocity was 1, the whole refiftance is to its part proper^
tional to Uie fquare of the velocity as 13^ + 48^ or 6l|^
to 48^ ; and Uierefore the refiftance of the pendulum in
water is to that part of the refifi;ance in air, which is propor*.
tional to the fquare of the velocity, and which in f^ft mo*
tioQS is the only part that deferves confideration, as 6\{^ to
48^ and 535 to Ij- conjunctly, that is, as 571 to 1; If the
whole thread of the pendulum ofcillating in the water had
been immcrfed, ita refiftance would have been ftill greater;
fo that the refiftance of the pendulum ofcillating in the water^
that is, that part which is proportional to the fquare of the ve«
locity, and which only needs to be confidered in fwift bodies^
it to the refiftance of the fame whole pendulum, ofcillating in
air with ifaefame velocity, as about 850 to 1, that is, as the
denfity of water to the denfity of air, nearly.
In this calculation we ought alfo to have taken in that
paort of the'vefifiaince of the pendulum in the water which was
as the fqnave of the velocity ; but I found (which will per-
haps ieem firange) that the refiftance in the water was aug«
mented in moie dian a duplicate ratio of the velocity. lot
ftarching after the caufe, 1 thought tepotk this, that the veflel
was too narrow for the magnitude of the pendulous globe, and
by its narrownefii obflruded the motion of the water as it
jridded to Ae ofcillating globe. For when I immerfed a pen-
daknis globe^ wliofe diameter was one inch only, the refift*
anoe was augmented nearly in a duplicate ratio of the velo-
ci^« I tried tUa by making a pendulum of two globes, of
wfakfa the lefler and lower ofcillated in the water, and the
greater and higher was fattened to the thread juft above the
02
84 MATHEMATICAL PKINCIPLC8 3ookll.
water^ and^ by ofciliating in the air^ affifted the motion of the
pendulum^ and continued it longer. The experiments made
bj this contrivance proved according to the following table.
Arc defer, in Jirji def cent . . 16 . 8 • 4 . 12 • 1 • ^ .^
Arc defer, in laji afcent . . 12 • 6 . S • 1^ • I- . •(• . -^
Pif.ofares,proport.to\ . o i 4 4. 4 j.
. motton loft J ^ ^ , ▼ ^
Number of ofeitlations . . . S| • &)• .n^^]|.S4.53.6$|
In comparing the reliftances of the mediums with each
other^ I alfo caufed ircm pendulums to ofciliate in qnickfilver;
The length of the iron wire was about 3 feet, and the dia-
meter of the pendulous globe about \ of an inch. To the wiie^
jnft above the quickfilver^ there was iixed another leadeir
globe of a bighe£s fufficient to continue the motion o( tb6
pendulum for fome time. Then a velTel^ that would faolft
^bout 3 pounds of quickfilver^ was filled by turns with qoidc^
filver and common water^ that, by making the pendnlambfr
cillate fucceffively in thefe two different fluids, I might find
the proportion of their refiftaaoes ; and the refiftanoe of the
quickfilver proved to be to the refiftance of water aib abott
13 or 14 to 1 ; that is, as the denfity of quickfilver to the den-
lity of water. When I made ufe of a pendulous globe fonie-
ihing bigger^ as of one whofe diameter was about |. or i^ of an
inch, the refiitance of the quickfilver proved to; be to the re-
fiftance of the water as about 12 or 10 to 1. But the fpnner
experiment is more to be relied, on, becaufe in the latter die
veflel was too.nartow in proportion to the magnitcde of the
immerfed globe ; for the veflel ought to have be^n enlarged
together with the globe. I intended to have repeated thefe
experiments with larger veflels> and in melted metals, and
other liquors both cold and hot ; but I had not Idfme to try
all ; and befides, from what is already defcribed, it appeain
fufficiently that the refiftance of bodies moving fwifUy is neady
proportional to the denfides of the fluids in which they move.
I do not fay accurately; for more tenacious fluids, of eqnai
denfity, will undoubtedly refifi; more than thofe thai aie notoie
liquid; as cold oil more than warm, warm oil more than iraiiii-'
iBfrater^ ^nd water more than fpirit of wine. But in liquor^
SeS. VL OM NATURAL PHILOSOPHV. g^
whicii are fenfibly fluid enongh^ as in air, in fait and frefli
water, in fpirit of wine, of turpentine, md falts, in oil cleared
of its faeces by diftillation and warmed, in oil of vitriol, and in
mercury, and melted metals, and any other fuch like, that
are fluid enough to retain for fome time the motion imprefled
upon them by the agitation of the vBflel, and which being
poured out are eafijy refolved into drops, I doubt not but the
•rule already laid down may be accurate enough, efpecially if
the experiments be made with larger pendulous bodies, and
more fwiftly moved.
Laflly, iince it is the opinion of fome that there is a cer-
tain aethereal medium extremely rare and fubtile, which freely
pervades the pores of all bodies ; and from fuch a medium, fo
^rvading the pores of bodies, fome refiftance muft needs
furife ; in order to try whether the refiftance, which we expe»
rknce in bodies in motion, be made upon their outward fuper*
ficies only, or whether their internal parts meet with any con-
iiderable refiftance upon their fuperficies, I thought of the
following experiment. I fufpended a round deal box by a
thread 1 J feet long, on a fi;eel hook, by means of a ring of the
fame metal, fo as to make a pendulum of the ^forefaid length.
The hook had a fliarp hollow edge on its upper part, fb that
the upper arc of the ring prefling on ihje edge might move the
ttiore fi^ly ; and the thread was fafi^ned to the lower arc of
"tiie ring. The pendulum being thus prepared, I drew it afide
from the perpendicular to the diftance of about 6 feet, and
that in a plane perpendicular to the edge of the hook, lefl; the
ringi while the pendulum ofcillated, ihouldflide to and fro on
the edge of the hook : for the point of fufpenfion, in which
the ring touches the hook, ought to remain immovable. I
•iberefbre accurately noted the place to which the pendulum
■was brought, and, letting, it go, 1 marked three other places,
to which it returned at the end of the 1ft, ^d, and 3d ofcil*
lation. This I often repeated, that I might find tbofe places
as accurately as pofiible. Then I filled the box with lead and
olher heavy metals that were near at hand. But, firft, I
weighed the box when empty, and tl^at part of the thread
that went round it, an4 half the remaining part^ extended
as
\
86 MATHEMATICAL PRINCIPLBft Book IL
between the book and the fufpended box ; for the thread fo
extended always ads upon the pendulum^ when drawn afide
from the perpendicular^ with half its weight. To this weight
I added the weight of the air contained in the boj^. And this
whole weight was about -^ of the weight of the box when
filled with the metals. Then becaufe the box when full of the
metals^ by extending the thread with its weighty increafed
the length of the pendulum^ I fhortened the thread fo at to
make the length of the pendulum^ when ofcillatihg^ tht fatne
as before. Then drawing afide the pendulum to the place firft
marked^ and letting it go^ I reckoned about 77 ofcillations
before the box returned to the fecond mark^ and as many aB*
terwards before it came to the third mark, and as many after
that before it came to the fourth mark. From whence I
conclude that the whole refiftance of the box, when full, had
not a greater proportion to the refiftance of the box, wh»
empty, than 78 to 77* For if their refiftances wece equals
the box, when full, by reafon of its vis infita, which was 78
times greater than the ^t^ infita of the fame when empty, ought*
to have continued its ofciilating motion fo much the longer,
and therefore to have returned to thofe marks at the end of
78 ofcillations. But it returned to them at the end of 77 of-
cillations.
Let, therefore, A reprefent the refiftance of the box upon its
external fuperficies, and B the refiftance of the empty box oa
its internal fuperficies ; and if the refiftances to the internal
parts of bodies equally fwift be as the matter, or the number
of particles that are refifted, then 78B will be the refift;ance
made to the internal parts of the box, when full ; and there-
fore the whole refiflance A + B of the empty box will be to
the whole refiftance A + 78B of the full box as 77 to 7B> and,
by divifion, A + B to 77B as 77 to 1 ; and thence A -)- B
to B as 77 X 77 to 1, and, by divifion again, A to B as
59^18 to 1. Therefore the refiftance of the empty box in' itii
internal parts will be above 5000 times lefs than the refiftance
on its external fuperficies. This reafoning depends upou the
fuppofition that the greater refiftance of the full box arifei
USe&.VJ. OF NATURAL PHILOSOPHY. 87a
Miot from any other latent caufe, but only from the tidlton of ■
Kfome fubtile fluid upon the included metal. I
Ht This experiment is related by memory, the paper being 1
mbtR. in which I had defcribed it ; To that I have been obliged
H|o omit forae fradtional parts, which are flipt out of my me-
^biory ; and I have no leifure to try it again. The lirft time
Bt made it, the hook being weak, the full box was retarded
Bboner. The caufe I found to be, that the hook was not
B|ht>ng enough to bear the weight of the box; fo that, as it
Hilcillated to and fro, the hook was bent fometimes this and
Hpinetimes that way. I therefore procured a hook of fuffi-
^bient ftrenglhj fo that the point of fufpenfion might remain
Hanmovedj and then all things happened as is above defcribed.
K SECTION vir.
WOfthe motion ofjluids, and the refiftance made to projeilcd
^t bodies.
K PROPOSITION XXXII. THEOREM XXVI.
^mVJ^ofe two fimilar fyfiems of bodies confijiing of an equal
K number of particles, and let the correjpondent particles be
H fimilar and proportional, each inonefyjtem to each in the
■- other, and have a like Jitualion among themfelves, and the fame
given ratio of denfity to each other; and let them begin to
move among themfelves in proportional times, and with like
motions (that is, thofe in onefji/iem among one another, and
thofe in the other among one another). And if the particles
that are in the fame fyjlem do not touch one another, except
in the moments of rejiexion ; nor attraS, nor repel each
other, except with accelerative forces that are as the diame-
ters of the eorrefpondent particles inverfely, and thefquares
of the velocities direSly; 1 fay, that the particles of thofe
fyfl^ems will continue to move among tltemfelvet with like mo-
tions and in proportional times.
Like bodies in like fituations are fuid to he moved among
themfelves with like motions and in proportional times, when
tbeir fituations at the end of thofe times are always found
2 in refpeft of each other: as fuppofe we compare the
irliclesin one fyftemwith the eorrefpondent particles in the
I Ueace tbe times will be proportional^ in which fijni-
G 4
88 MATftEMATtCAL PftlNClPLEB jBooft tl«.
lar and prbjportional parts of fimilar figures will be tlefcribed
by correfpondent • particles. Therefore if we fuppofe two
fyftems of this kind> the correfpondent particles, by reafon of
the fimilitode of the motions at their beginning, will continue
to be moved tvitb like motions^ fo long as they move wttb-
oat meeting one another ; for if they are aded OB by 'no
forces, they will go on uniformly in right lines^ by the ift
law. But if they do agitate one another with fotne 4:ertahi
forces, and thbfls forces are iis the diameters of the cdrreljpmMl-
ent particles inverfely and the fqiiares of the velocities diiefti'
ly, then^ becanfe the particles are in like fituations^ and tfaek
forces are proportiokial^ the whole forces with which can^
fpondent particles are agitated, and which areoompoonded of
each of the agitating forces (by torol. £ of the laws), will
have like directions, and have tbefame effeA as if they r^
fpeded centres placed alike among the particles; and thofe
whole forces w31 • be to each other as the feveral forces
which compofe them, that is^ as the diameters of the ootre-
fpondent particles inverfely, and the fqaares of the velocities
dire<^ly: and* therefore will caufe correfpondent patticks to
continue to deibribe like figures. Thefe things will be fo
(by cor. 1 and 8, prop. 4, book I)/ if thofe centres areatrefi;
but if they are moved, yet> by reafon of the fimilitude of
the tranflations,' their fituations among the partides of the
fyftem will remain fimilar; fo that the changes itaitroduced
into the figures defcribed by the particles will HAM be fimilar.
So that the motions of cori'efpondent and fimilar particles
vrill continue fimilar till their firft meeting with each other;
and thence will arife fimilar coUifions, and fimilar reflexions;
which will again beget fimilar motions of the particles among
themfelves (by what was juft now- (hewn), till they mntuslly
fall upon one another again, and Co on ad infinitum.
CoR. 1. Hence if any two bodies, which are fimilar and
in like fituations to the correfpondent particles of die fyflems,
begin to move amongft them in like manner and in propor-
tional times, and thedr magnitudes and denfities be -to eadi
other as the magnitudes and denfities of the cottefponding
-particles, thefe bodiiM will edntitmelo be^mioved in like
SeA. VIL OF NATI]ftAL PHILOSOPHY. 89
<ner and in proportional times; for the cafe of the greater
-part§'of both fyftems and of the particles is the very fame.
Cor; 2. And if all the fimilar and fimilarly fituated parts
of both fyftems be at reft among themfelves; and two of them^
'^hich are greater than the reft^ and mutually correfpondent
in both fyftems^ begin to move in lines alike pofited^ with
any fimilar motion whatfoever^ they will excite fimilar mo*
lions in the reft of the parts of the fyftems^ and will continue
to move among thofe parts in like manner and in proportion^
times; and will therefore defcribe fpaces proportional to their
diameters.
PROPOSITION XXXIII. THEOREM XXVII.
'I%t fsuike things being fuppofed, I fay, that the greater parti
of the fyftems arc refijied in a ratio compounded of the
duplicate ratio of their velocities, and the duplicate ratio
of their diameters, and thejimple ratio of the denfrty of the
'parts of the fyftems.
For the rdiftance arifes partly from the centripetal or
centrifugal forces with which the particles of the fyftem
mutually aft on each other^ partly from the coUifions and
reflexions of the particles and the greater parts. The refift*
ances of the firft kind are to each other as the whole motive
forces from which they arife, that is^ as the whole accelera*"
live forces and the quantities of matter in correfponding
•parts; that is (by the fuppofition)^ as the fquares of the velo-
cities dire6Uy, and the diftances of the correfponding particles
inverfely^ and the quantities of matter in the correfpondent
parts direftly ; and therefore iince the dift;ances of the parti-
cles in one fyftem are to the correfpondent diftances of the
particles of the other as the diameter of one particle or part
in the former fyftem to the diameter of the correfpondent
particle kx part in the other^ and fioce the quantities of mat^
ter are as' die denfities of the parts and the cubes of the dia^
meters; the refiftances are to each other as the fquares of
- the velocities and the fquares of the diameters and the den-
fities of the parts of the fyft:ems. Q.E.D. The refiftances df
the latter fort are as the number of correfpondent reflexions
and the forces of thofe reflexions conjun Aly ; but the number
^ MATftSMATlCAL PEINCIPLBS JSoOJfc.H.
•of the reflexions are to each other as the velocities of the cor-
refponding parts diredUy and the fpaces between their reflex*
ions inverfelj. And the forces of the reflejcions are as the
Telocilies and the magnitudes and the deniities of the corre-
iponding parts conjandUy ; that is^ as the velocities and the
cubes of the diameters and the denfities of the parts. And^
joining all thefe ratios, the reflftances of the oorrefponding
parts are to each other as the fquares of the velocities and
ihe fquares of the diameters and the denfities of the ^parts
1X>xyan6Uy. Q.E.D.
Cor. 1. Therefore if thofe fyftems are two elafluc fluids,
like our air^ and their parts are at reft among themfelves; and
two fimilar bodies proportional in magnitude and denfitjr to
the parts of the fluids^ and iiitiilarly fituated among thofe
partSj be any how proje&ed in the direiflion of lines fimilarly
pofited; and the accelerative forces with which the particles
of the fluids mutually a6l upon each other are as the diame-
ters of the bodies projedled inverfely and the fquares of their
yelocities dire^y; thofe bodies will excite fimilar moticms in
the fluids in proportional times> and will defcribe fimilar
fpaces and proportional to their diameters.
Cob. 2. Therefore in the fame fluid a projedted body that
moves fwiftly meets with a reflftance that is in the duplicate
latio of its velocity^ nearly. For if the forces, with which
4iftant particles a^ mutually upon one another fliould be
augmented in the duplicate ratio of the velocity, the prqje^
ed body would be refifl^d in the fame duplicate ratio ac-
curately ; and therefore in a medium, whofe parts when at
a difiance do not a6); mutually with any force on one anothen
the refiftance is in the duplicate ratio of the velocity accu-
rately. Let there be, therefore, three mediums A, @, C, con-
fifiiog of flmilar and equal parts regularly difpofed at equal
difl;ances. Let. the parts of the mediums A and B recede
from each other with forces that are among themfelves aa T
and V ; and let the parts of the medium C be entirdy defli-
itute of any fucfa forces. And if four equal bodies J), E, F, G^
moie in thefe mediums, the two firft D and £ in the two firft
A and B^ and the other two F and.Gin the third C^ and if
Sc£i. Vn. OF NATDBAL PHILOSOPHY. 01
the velocity of the body D be to the velocity of the body E,
and the velocity of the body F to the velocity of the body G,
in the fubduplicate ratio of the force T to the force V; the
reGftance of the body D to the refiftance of the body E, and
the reGJlance of the body F to the refiftance of the body G,
will be in the duplicate ratio of the velocities ; and therefore
the refiftance of the body D will be to the refiftance of the
body F as the refiftance of the body E to the refiftance of the
body G. Let the bodies 1) and F be equally fwift, as alfo the bo-
dies E and G ; and, augmenting the velocities of the bodies
D and F in any ratio, and diminilhing the forces of the
particles of the medium B in the duplicate of the fame
ratio, the medium B will approach to the form and condition
of the medium C at pleafure;and therefore the refiftances of
theeqnaland equally fwift bodiesE and Gin thefe mediums will
perpetually approach to equahty, fothat their difference will at
laftbecomelefsthananygiven. Therefore,fincetherefiftancesof
the bodies D and F are to each other as the refiftances of the bo-
dies E and G, thofe will alfo in like manner approach to the
ratio of equality. Therefore the bodies D and F, when they
move with very great fwiftnefs, meet with refiftances very
nearly equal ; and therefore fince the refiftance of the body
F is in a duplicate ratio of the velocity, the refiftance of the
body D will be nearly in the fame ratio.
Cob. 3. The refiftance of a body moving very fwifl in an
eJaftic fluid is almoft the fame as if the parts of the fluid were
deftitute of their centrifugal forces, and did not fly from each
other; if fo be that the elafticity of the fluid arife from the
centrifugal forces of the particles, and the velocity be fo
great aa not to allow the particles time enough to ai5i.
Cor. 4. Therefore, fince the refiftances of fimilar and
equally fwift bodies, in a medium whofe diftant parts do not
fly from each other, are as the fquares of the diameters, the
refiftances made to bodies moving with very great and equal
velocities in an elaftic fluid will be as the fquares of the dia-
meters, nearly.
Cor. 5. And fince fimilar, equal, and equally fwift bodies,
moving through mediums of the fame denflty^ wbofe parti-
pt . .MATRSMATICAX 1»1tlNCIPLES .Sdok II.
^(deg do'Dot fly from each other mutually^ will ftrike i^inft an
.fiqvftl quantity of matter in equal times, whether the partides
^ which the medium coufifls be more and fmaller, or fewer
apd greater^ and therefore imprefs on that matter an equal
^quantity of motion^ and in return (by theSd law of motion)
' iuffnrao equal^ re-«iSion from the fame> that is^ are equally re-
. iiited.; it is manifeft, alfo^ that in elaftic fluids. of the fame den-
fity, when the bodies move with extreme fwiflnefs, their re-
fiftances are nearly equals whether the fluids confifl: of grofi
parJtSj or of parts ever fo fubtile. For the reflftance of pro-
je^es moving with exceedingly great celerities is not much
diminiflied by the fubtilty of the medium.
. CoR. 6. All thefe tbines are fo in fluids whofe elailic. force
takes its rife from the centrifugal forces of the particles.
But if that force arife from fome other caufe^ as from theexr
paixfiipn of the particles after the manner of wool, or the
boughs of trees^ or any other caufej by which the partides
me hindered from moving freely among themfelves, the refifl:-
^riccj by reafon of the lefler fluidity of the medium^ will be
^eater than in the corollaries above.
. PROPOSITION XXXIV^ THEOREM XXVIII.
If. in a raremedium, confifiing.of equal particles freely difpofed
. . fit fqual diftancesfrom each other, a globe and a a/Under de-
fcribed on equal diftmeters move with equal velocities in the
dire&ion of the axis of th^ cylinder, the rejiftance of the
globe will H but halffo great as that, of the cylinder.
. For fince the a&ion of the medium upon the body is the
famie (by cor. 5 of the laws) whether the body move in a qui^
efcent medium^ or whether the particles of the medium im*
pinge with the fame velocity upon the quiefoent bodyy let us
confider the body us if it were quiefcent^ and fee .with what
f^rce it would be, impelled by the moving medium. JiCt,
therefore, ABKI (PJ. 6, Fig. 9) peprefent a fpherical body de*
fcribed from the c^tre C with the femi-diameter CA> and
let the.partides of the medium impinge with a given velocity
upon that fpherical body in the dire6lions of right Mncs. pa-
ra}lf^ to AC; and let FB be one of thofe right lines. . In FB
^k^.J3 equal to^he. feoM-dic^meter CB^ wd draw BD toucbr
SeS. Vn. OV NATURAL PHTLdSOPHT. 9$
ing the i|)here ia B. ; Upbti KC and BDietfall the peif^en*.
dicalars BE> LD ; and the force with which a particle otiibk
mediutiif impinging on the /globe obliquely ifa the^diredion
FB, Would ftrike the globe an B, will be to the foree: With
which the fame particle^ meeting the cylinder ONGQ d(^.
fcribed' about the globe with tbb axis ACl, wotild fiiike it |>i^p-
pendictlaifjy in b, as LB to LB/ or BE to BC. ii^aite; the
e&cs^ty of this force to move the globe according io theidif;
redHoniof its incidence FB or AC, is to the efficacy of: the
fame tomove the globe according to the diredlion of itsd^fer*
mination, that is, in . the diredion of the.rigbt lineBCI iiii
whiohMtiim^els the globe dire&iy,:'as BE to BC* Aim}) joi^
ing thefe Yatios, the efficaty of a particle; felling '^pojftithid
globe ol^Equely in the dire^ioh of thd right line EB, tdimiiim
the globe in the dire6i;ion of ibs incidence is to th^ effidiid^
of tide fame. pai*ticle falling m the fauie iine:ptrpendi)6ukd§[
on the cylinder, to move it in the fame ^ire^ion, as BBl itSi
BC\> . Therefore if in bE, which is perpendicular, to tbdjcira
cular bale of the cylinder NAO, and equal to the radiiis AlC!>
■■■. -■: ' \t^- • BE* . ■ • ■ "^ • J' Iii
we take bH equal to -7:^= 9 then bH will be to bE as the eF-
feA of the particle upon the globe to tfaeefifedl of tliietpeir*^
tide upon the cylinder. And therefore the foM whiicbi is
fbitned by all the right lines bH will be to the folid formed:igr
all tie ii^t lines bE as the eSe€i of all thie particles. u|idnst(he
globe to the effe& 6f all the particles upon the cyliqdsjl) Butt
the formier- of thefe folids it a paraboloid whpfe vertex 10 6y
its axis' CA, and latus re6ium CA, and the latter folid 'i^a>cj[>#
linder circumfcribing the paraboloid ; .and it is known tbatici
paraboloid is half its cirdumfcHbed cylinder. Therefore tho
wholeiforceof the medium upon the globe is half of the en-
tire force of the fame upon the cylinder. And therefore iC
the particles of the medium are at reft^ and the cylinder and
globe move with equal velocities, the refiftance of the globe
will be half the reOftance of the cylinder. Q.E.D.
SCHOLIUM.
By the fanie method other figures may be compared to-
gether as to then: refiftance ; and thofe may be found which
94 If ATHXMATICAL PBINCIPLES Book 11^
are moft apt to continue their motions in refifting mediunup.
As if upon the circular bafe CEBH (PI. 6^ Fig. 3) from the
centre 0> with the radius OC» and the altitude OD, cme
would conftrudl a fruftum CBGF of a cone, which ihould
meet with lefs refiftance than any. other fruftum conftruAed
with the fame bafe and altitude, and going forwards towards
D in the direction of its axis: bifeA the altitude OD in Q,
and produce OQ to S, fo that QS may be equal to QC, and
S wiU be the vertex of the cone whofe fruftum is fought.
- Whence, by the bye, fince the angle CSB is always acate,
h foUows, that, if the folid ADBE (PI. 6, Fig. 4) be generated '
by the convolution of an elliptical or oval figure ADBE aboai
its axis AB, and the generating figure be touched by three
right lines F6, 6H, HI, in the pointa F, B, and I, fo that
6H fhall be perpendicular to the axis in the point of contaft
B, and F6, HI may be inclined to GH in the angles FGB>
BHI of 135 degrees; the folid arifing from the convolatioii
of the figure ADF6HIE about the fame axis AB will be lefii
xefifl^d than the former folid ; If -fo be that both move forward
in the dire&ion of their axis AB, and that the extremity B o^
each go foremoft. Which propofition I conceive may be of
nfe in the building of (hips.
If the figure DNF6 be fuch a curve, that if, from any
pomt thereof, as N, the perpendicular NM be let fall on the
. axis AB, and from the given point 6 there be drawn thei%ht
line GR parallel to a right line touching the figure in N, and
cutting the axis produced in R, MN becomes to GR as GR'
to 4BR X GBS the folid defcribed by the revolution o^
this figure about its axis AB, moving in the before*mentioned
rare medium from A towards B, will be lefs refifted than any
other circular folid whatfoever, defcribed of the fame leng^th
and breadth.
f:^ TkedemoHjhaiionefihefe curious Theoremt heifig omiiied iy the auih^^ ikw
iuaigfis thereof ^ ce mrnumea ied hy a friend j is added at the end of this voiume.
PROPOSITION XXXV. PROBLEM VII.
If a rare medium confiji ofveryfmall quiefcent particles of
equal magnitudes, and freely difpofed at equal diftancesfram
FL^/r FI. m.F.
m;-
P^^S4
ic>r a
M^:^.
C O rt^D
L. F
F^.'S,
■ R
Se6i. VII. OF NATURAL PHILOSOPHY.
one another ; to find the refiftanct of a globe moving
Jormty foTwardi in ihis medium.
Case 1. Let a cylinder defcribed with the fame diameter
and altitude be conceived to go forward with the fame velo-
city iQ the direftion of its axis through the fame medium ;
and let ns fuppofe that the particles of the medium, od which
the globe or cylinder falls, fly back with as great a force of re-
flexion as poflible. Then fince the refiftance of the globe (by
the laft propolition) is but half the refiftance of the cylinder,
and fince the globe is to the cylinder as 2 to 3, and fince the
cyhnder by falling perpendicularly on the particles, and re-
flecting them with the ulmoll force, communicates to them a
velocity double to its own ; it follows that the cylinder, in
moving forward uniformly half the length of its axis, will com-
municate a motion to the particles which is to the whole
motion of the cyhnder as the denfity of the medium to the
denfity of the cylinder; and that the globe, in the time it de-
fcribes one length of its diameter in moving uniformly for-
wards, will communicate the fame motion to the particles;
and in the time that it defcribes two thirds of its diameter, will
communicate a motion to the particles which is to the whole
motion of the globe as the denfity of the medium to the den-J
fity of the globe. And therefore the globe meets with a re-
iiftance, which is to the force by which its whole motion may
be either taken away or generated in the time in which it de-
fcribes two thirds of its diameter movina; uniformly forwards,
as the denfity of the medium to the denfily of the globe.
Case 2. Let us fuppofe thiit the particles of the medium
incident on the globe or cylinder are not reflefled ; and then
the cylinder falling perpendicularly on the particles will com-
municate its own fimple velocity to them, and therefore meets
a refiftance but half fo great as in the former cafe, and the
globe alfomeeta with a refiftance but half fo great.
Case 3. Let us fuppofe the particles of the mediui
back from the globe with a force which is neither the greate^
nor yet none at all, but with a certain mean force; then the'
refiftance of the globe will be in the fame mean ratio between
96^ MATHBMATICAL PRINCIPIBS JBMJb H;.
the refiftance m the fiiit cafe aod the refiftance in the fe-
cond. Q.E.I.
Co&. 1. Hence if the globe and the particles are infinitelj
hard^ and deftitate of all elaftic force, and therefore :of all
force of reflexion ; the refiftance of the globe will bei to the
force by which its whole motion may be deftroyed or gene*,
ratedj in the time that the globe defcribes four third parts .of
its diameter, as the denfity of the medinm to the d^ifity of
tlie globe.
CoR. 2. The refiftance of the globe, caUrii paribmi, is in:
the duplicate ratio of the velocity.
CoR. 3. The refiftance of the globe, utteri$paribfi9,.iBki^
the duplicate ratio of the diameter.
Cor. 4. The refifl^nce of the globe is, cmteru parUnu^ as
the denfity of the medium.
CoR. 5. The refiftance of the globe is in a ratio compoonded
of the duplicate ratio of the velocity, and the duplicul^ nitiO)
of the diameter, and the ratio of the denfity of the; medium.
Cor. 6. The motion of the globe and its refiflanoe may be,
thvip expounded. Let A B (PL 7> Fig. 1) be the time in which:
the globe may, by its refiftance uniformly continue^^ lofe its
whole motion. Eredl AD, BC perpendicular to AB« Ijei
BC be that whole motion, and through the point Q the
afymptotes being AD, AB, defcribe the hyperbola CF. Pro-,
duce AB to any point £. £re& the perpendicular ^F meet*,
ing the hyperbola in F. Complete the parallelograiii . CBEG^
and draw- AF meetipg BC in H. Then if the globe;in any;
time BE, with its fiirft motion BC uniformly continued, de>
fcribes in a non-refifting medium the fpace CBEG expoondi^;
by the area of the parallelogram, the fame in a refifting me*;
dium will defcribe the fpace CBEF expounded by the area <^.
the hyperbola ; and its motion at the end of that time will be.
expounded by £F, the ordinate of the hyperbola, there, being-
loft of its motion the part FG. And its refifi:ance at the eiid
of the fame time will be expounded by the length BH, there;
being loft of its refifl^ance die part CH. All thefe things ajK
pear by cor. 1 and 3, prop. 5, book 12.
Se3. Vn« O? NATUKA& PHIIi^SOJ^HY. 97
Cor. 7. Hence if the globe in the time T hy the re^ilance
{I Uttifermly continued lofe its whole motion M^ the feme
£lobe in the time t in a reiifting medium^ wherein the refifl>
jupce R dpcreafea in a duplicate ratio of the velocity^ will lofe
. ,^ , tM , TM
out of Its motion M the part Tfr-r-.J the part ,«—— remaining;
wd will defcribe a fpace which is to the fpace defcribed in the
i^me time t> with the uniform motion M^ as the logarithm of
T + t
the number — ss — multiplied by the number 2,302585092994
is to &e number 7^,9 becaufe the hyperbolic area BCFE is to
the redUagle BCGE in that proportion.
SCHOLIUM.
1 have exhibited in this proportion the re&ftance and re-
taidation of fpherical projediles in mediums that are not con*
tinned 9 and (hewn that this refiftance is to the force by which
the whole motion of the globe may be deftroyed or produced
in the time in which the globe can defcribe two thirds of its
diameter, with a velocity uniformly continued, as the denfity
of the medium to the denfity of the globe, if fo be the globe
i^d the particles of the medium be perfe<^ly elaftic, and are
endued with the utmoft force of reflexion ; and that this force,
where the globe and particles of the medium are infinitely
hard and void of any refle&ing force, is diminifhed one half.
Bat in continued mediums, as water, hot oil, and quickfilver,
the g^obeas it pafies thmugh them does not immediately
firike againft all the particles of the fluid that generate the re-
fiftance made to it, but prefles only the particles that lie next
to it, which prefs the particles beyond, which prefs other par-
ticles, and fo on ; and in thefe mediums the refifi:ance is di*
mtniibed one other half. A globe in thefe extremely fluid
medinmg meets with a refiftance that is to the force by which
its whole motio^ may be defl;royed or generated in the time
wherein it can defcribe, with that motion uniformly continued,
eight third parts of its diameter, as the denfity of the men
dium to the denfity of the globe. This I ihaU endeavour to
(hew in what follows.
Vol. n. H
98 MATHEMATICAL PRIKC1PLB8 " Book II*
PROPOSITION XXXVI. PROBLEM VIII.
To define the motion of water running out of a cylindrical
vejfel through a hole made at the bottom.
Let ACDB (PI. 7> Fig. 2) be a cylindrical veflel, AB tbe
mouth of it, CD the bottom parallel to the horizon, £F a
circular hole in the middle of the bottom, G the centre of tbe
hole, and GH the axis of the cylinder perpendicular to the
horizon. And fuppofe a cylinder of ice APQB to*b« of tbe
fame breadth with the cavity of the veffel, and to have ihft
fame axis, and to defcend perpetually with an uniform mo<^
tion, and that its parts, as foon as they touch the fuperficiea
AB, difTolve into water, and flow down by their weight into
the vefTel, and in their fall compofe the catara&or oolumn o£
water ABNFEM, paifing tHrough the hole EF, and filling up
the fame exactly. Let the uniform velocity of thedefcending
ice and of the contiguous water in the circle AB be that which
the water would acquire by falling through the fpace IH ; and
let IH and HG lie in- the fame right line; and through the
point I let there be drawn the right line KL parallel to tbe
horizon, and meeting the ice on both the fides thereof in JL
and L. Then the velocity of the water running out at the
hole EF will be the fame that it would acquire by fklling
from I through the fpace IG. Therefore, by GaKlco^% the-
orems, IG will be to IH in the duplicate ratio of the velocity
of tbe water that runs out at the hole to the velocity of the
vfater in the circle AB, that is, in the duplicate ratio of th^
circle AB to the circle EF ; thofe circles being reciprocally as
the velocities of tbe water which in the fame time and in equal
quantities paffes feverally through each of them, and com-
pletely fills them both. . We are noyr confidering the ydocity
with which the water tends to the plane of the horizon. Bat
the motion parallel to the fame, by which the parts of the
filing water approach to each Ptheo is not \^^xe taken notice
of; fince it is neither produced by gravity, npr ^t all changes
the motion perpendicular to the horizon which the gravity
produces. We fuppofe, indeed, that the parts of the water
cohjerie a little, that by their cohefion they may ip fallipg ap-
proach to each other with motions parallel to (be horizpp. in .
iSiS. Vil. OP NATtRAL PHILOSOPHY. 99
order to form one fingle caiarad);^ and to prevent their being
divided into feveral: but the motion parallel to the horizon
arifing from this cohefion does not come under our prefent
Gonfideration.
Casb 1. Conceive now the whole cavity in the veffel,
which encompafles the falling water ABNFEM^ to be full of
ice^ fb that the water may pafs through the ice as through
a funnel. Then if the water pafs very near to the ice only,
without touching it; or^ which is the fame thing, if, by rea-
fon of the perfedl fmoothnefs of the furface of the ice, the
water, through touching it, ghdes over it with the utmoft
freedom, and without the leaf): refiilance ; the water will run
through the hole'EF with the fame velocity as before, and
the whole weight of the column of water ABNFEM will be
all taken up as before in forcing out the water, and the bot-
tom of the veiTel will fiiftain the weight of the ice encom-
pafiBng that column.
Let now the ice in the veffel diffolve into water; yet wiH
the efflux of the water remain, as to its velocity, the fame as
before. It will not be lefs, becaufe the ice now diflblved wiH
endeavoor to defcend.; it will not be greater, becaufe the ice^
now become water, cannot defcend without hindering the
defcent of other water eqnal to its own defcent. The fame
force ought always to generate the fame velocity in the efflu-
ent water.
Bat the hole at the bottom of the veffel, by reafon of the
oblique niotions of the particles of the effluent water, mufl be
a little greater than before. For now the particles of the
water do not all of them pafs through the hole perpendicu-
larly, but, flowing down on all parts from the fides of the veflel,
and converging towards the hole, pafs through it with oblique
motions; and in tending downwards meet in a fiream whofe
diameter is a little fmaller below the hole than at the hole
jtfelf ; its diameter being to the diameter of the hole as 5 to 6,
or as 5f to 6|, very nearly, if I took the meafures of thofe dia-
meters right. I procured a very thin flat plate, having a hole
pieirced in the middle, the diameter of the circular hole being
^ parts of an inch. And that the {Iream of running waters
100 MATHfiMATlCAL PlUKCXBLK^ Book U^
might not be accelerated in feUtng, and hj that acccleraiiott
become narrower, I fixed this plate not to the bottom^ ba^
to the fide of the veiTel, fo us to make the water go oat m the
dire<flion of a line parallel to the horizon. Then^vhen tho
vefTel was full of water, I opened the hole to kt it nm oat;
and the diameter of the ftream, meatiired with great accuiafijr
at the diftance of about half an iqch from the hole,* was ^ ofi
an inch. Therefore the diameter of tbii circular hole wat.to
the diameter of tlie ftream very nearly as 26 to fil. So that
the water in pailing through the hole converges on all fidca^
and, after it has run out of the veiTel, becomes fmaller by coii^
verging in that manner, and by becoming fmaller is accelecatn
ed till it comes to the diftance of half an inch from the hcde,
and at that diftance flows in a fmaller ftream and with greatcs
celerity than in the hole itfetf, and this in the ratio of £5 X*
9^ to 21 X 21, or 17 to IQ, very nearly ; that is, in about the
fubduplicate ratio of £ to 1 . Now it is certain from experi-^
ments, that the qoantity of water nmning out in a givea
time through a circular hole made in the bottom of ave&l
is equal to the quantity, which, ftowing with the.afofdaid
velocity, would run out in the fame time through another ciiw
cular hole, whofe diameter is to the diameter of the former
as 2 1 to 25. And therefore that running, water in paffiag
through the hole itfelf has a velocity downwards equal to that
which a heavy body would acquire in falling through half die
height of the ftagnant water in the vefiel, nearly* But, then,
after it has run out, it is fiitl accelerated hy converging^ till U
arrives at a diftance from the hole that is neailji equal to its
diameter, and acquires a velocity greater than the other in
about the fubduplicate ratio of £ to I; which velocity a heavy
body would nearly acquire by falling through tlie. wrholii
height of the ftagnant water in the vefi*eL
, Therefore in what follows let the diameter of Ifae ftream be
reprefented by that leiEeK hole which we called £F. And
imagine another plane Y W above, the hole EF (PL 7, Fig«
S), and parallel to the plane thereof, to be placed at a difiaooa
equal to the diameter of the ikme hole, and to be pieroe4
through with. a>grntec bole ST, of £och a magnitude t)ia|t
SiS. VII. t)F IfflTUKAL PHILOSOPHY. 101
a ftream which will ^^h€t\y fill the lower hole £F may paik
tbrougtb it; the diameter of which hole will tlierefore be to
the damieter of the lower hole as 25 to 21^ nearly. By this
zHeaDs the water will rua perpendicularly out at the lower
hoie ; «iid tiie quantity of the water running out will be^ ac-
cording to the magnitude of this laft hole^ the fame^ very
oeariy, wlitch the folntion of the problem requires. The
Spdibt included between the two planes and the faHUng ftream
may be con&dered as the bottom of the veflel. Bat^ to maike
4he fekitkm more fimple and mathematical^ it is better to take
'the lower plane alone for the bottom of the vei>el^ and to
Ittppofe that tfaes water wbich flowed through the ice as
4;broagh a fbmiel^ and ran out of theveiTel tlirough the hole EF
made in the lower plane, prefervesits motion continnaUy, and
ihat the ice continues at reft. Tbere&re in what follows let
ST be tbe diaflneter of a circnlar hole defcribed from the cen-
tre Z> and let the fiream run out of the veflel through that
hole> when th^ water in the vefiel is all fluid. And let £F be
the diameter t>f tbe hole, which tbe ftream, in £Uling through,
e«M^y fills up^ whether the water runs out of tbe veffid hy
that itpper bole ST> or flows through tbe middle of tlie ice in
tbe ^leflel, tis thrcragh a funnel. And let the diameter of tbe
Qfpeir iiole ST be to the diameter of the lowor EF as about U
to SI, and let the perpendicular di&tfnce betwaeen die planes
4lfihe faoiesbe eqoal to the diameter of the leiTerhole £F.
ThM tbe Tckeity of the water downwards^ in running out of the
▼faffiri through the hole ST, will be in that hole tfie ikme that
a -body nay jbcc)uire by falling from half the iieight IZ; and
the ¥docity of both the falUng ft reams will be in the hole £F>
the ffttee which a body woald acquire by faUing from* the
whde b^ght IG.
CiUis £. If the hole EF be not in tlie middle of the hbt*
took «^ thd Teffel^ bat in fcane other part thei*e«Bf^ tbe water
will ftiH tw^ out with the fimie vdocity as befooe, if the m^ig*
nitnde of the hole be the fame. For thon^ an hei|vy body
tabeis a longer time in defcending to tbe fame depth;^ 'iiy an
oblique Ime, than by a perpendicular line^ yet in both cafes
itrAcqaires in its defcoU ibe Anne velocity ; ^ G^lilcq fai^s
demonftrated. H 3
•• *
102 MATHEMATICAL PRINCTPtES Sook IT.
Case 3. The velocity of the water is the fame when it
ruQs out through a hole in the fide of the veflel. For if the
hole be finally fo that the interval between the fuperficies AB
and KLmay vanifh as to fenfe^ and the ftream of water hori^
zontally ifTuing out may form a parabolic figure : from the
latns re6lum of this parabola may be coIle6led^ that the ve)o«
city of the effluent water is that which a body may acquire by
falling the height IG or HG of the ftagnant water in the vet
fel. Foo by making an experiment, I found that if the height
of th^: ftagnant water above the hole were 20 inches, and the
height of the hole above a plane parallel to the horizon were
alfo 20 inches, a ftream of water fpringing out from thence
would fall upon the plane, at the diftance of 37 inches^ very
nearly, from a perpendicular let fall upon that plane from tlie
bole. For without refiftance the ftream would have fallen
upon the plane at the diftance of 40 inches, thehtii»rieA«a
of the parabolic ftream being 80 inches.
Case 4. If the effluent water tend upwards, rt w31 ftill
iflue forth with the fame velocity. For the fmall ftreami ci
water fpringing upwards, afcends with. a perpendicular hi6^
tion to GH or GI, the height of the ftagnant water in the
veflel; excepting in fo far as its afcent is hindered a litdehy
the refiftance of the air; and therefore it fprings out with fh^
fame velocity that it would acquire in falling from ihat heigh tr
Every particle of the ftagnant water is equally prefled on all
fides (by prop, ig, book 2), and, yielding to the preffnie,
tends all ways with an equal force, whether it defoendi
through the hole in the bottom of the veflel, or gaflies out
in an horizontal diredlion through an hole in the fide^ or
pafles into a canal^ and fprings up from thence through a lit-
tle hole made in the upper part of the canal. And it may
not only be colle6led from reafoning, but is maniieft alfo
from the well-known experiments juft mentioned, that the
velocity with which the w&ter runs out is the very fame that
is afligned in this propofition. '
Case 5. The velocity of the effluent water is the fame,
whether the figure of the hole be circular, or fquare, or tri«
wg^ht, pr any other figure equal to the circular ; for the ve*
3eS. VIL OP NATURAL PHILOSOPHY. 103
locity of the effluent water does not depend upon the figure of
the hole, but arifes from its depth below the plane KL.
Case 6. If the lower part of the veiTel ABDC be immeriV
ed iato ftagnant water, and the height of the ftagnant water
above the bottom of the veffel be GR, the velocity with
which the water that is in the veflel will run out at the hole
£F into the ftagnant water will be the fame which the
water would acquire by falling from the height IR; for the
weight of all the water in the veflel that is below the fuper^
ficies of the ftagnant water will be fuftained in equilibrio by
the weight of the ft;agnant water, and therefore does not at
all accelerate the motion of the defcending water in the vef-
fel. This cafe will alfo appear by experiments, meafuring
the times in which the water will run out.
^JoB. !•• Hence if CA the depth of the water be produced !i
to K, fo that AK may be to CK in the duplicate ratio of the ;
area of a hole made in any part of the bottom lo the area of
the circle AB, the velocity of the effluent water will be equal
to the velocity which the water would acquire by falling
from the height KC.
Cor. 12. And the force with which the whole motion of |
the effluent water may be generated is equal to the weight of
•cyliodric column of water, whofe bafe is the hole EF, and its
altitude SGI or £CK. For the effluent water> in^ the time it
becomes equal to this column, may acquire, by falling by its
own weight from the height GI, a velocity equal to that with
which it runs out.
€oB. 3. The weight of all the water in the veflel ABDC is
to that part of the weight which is employed in forcing out
the wal» as the fum of the ^*ircles.AB and EF to twice the
circle EF. For let 10 be a mean proportional between IH
and IG, and the water running out at the hole EF will, in
the time that a drop falling from I would defcribe the alti-
tude IG, become equal to a cylinder whofe bafe is the circle
EF and its altitude 2l6, that is, to a cylinder whofe bafe is
the circle AB, and whofe altitude i9 £I0. For the circle EF
is to the circle AB in the fubduplicate ratio of the altitirde
IH to the altitude IG ; that is, in the fimple ratio of the
H 4
106 MATHEMATICAL PRINCIPLES Book II«
hend within it the column of congealed water PHQ^ thc^
weight of which is fuftained by that little circle. For thoa^
the motion of the water tends diredUy downwards, the ex^*
temal fuperficies of that column mull yet meet the bafe PQ
in an angle fomewhat acute, becaufe the water in its fall it
perpetually accelerated, and by reafon of that acceleratbn
become narrower. Therefore, flnce that angle is lefk than •
right one, this cplumn in the lower parts thereof will lie
within the hemi-fpheroid. In the upper parts alfo it will be
acute or pointed; becaufe, to make it otherwife, the hori*
zontal motion of the water muft be at the vertex infinitely
more fwift than its motion towards the horizon. And the left
this circle PQ is, the more acute will the vertex of this column
be ; and the circle being diminiihed in infinitum, the angle
PHQ will be diminiihed in infinitum, and therefore the co*
Inmn will he within the hemi-fpheroid. Therefore that co«
lumn is lefs than that hemi-fpheroid, or than two third parts
of the cylinder whofe bafe is that little circle, and its altitude
GH. Now the little circle fuftains a force of water equal to'
the weight of this column, the weight of the ambient water
being employed in caufing its e£9ux out at the hole4
CoR. 9* The weight of water which the little circle PQ
fuftains, when it is very fmall, is very nearly equal to the
weight of a cylinder of water whofe bafe is that little circle^
and its altitude }GH; for this weight is an arithmetic^
mean between the weights of the. cone and the hemi-fpheroid
above-mentioned. But if that little circle be not very fnialV
but on the contrary increafed till it be equal to the hole £F^
it will fuftain the weight of all the water lying perpendicularly
Above it, that is, the weight of a cylinder of water whofe bafe
is that little circle, and its altitude GH«
' CoR. 10. And (as far as I can judge) the weight which this
little circle fuftains is always to the weight of a cylinder of
water whofe bafe is thatlittJe circle, and its altitude |GH, as
EF* to EP — iPQS or as the circle EF to the excels of this
circle abov^ half the little circle PQ, very nearly.
LEMMA IV.
If a cylinder move uniformly forwards in the direQion of its
length, the refinance made thereto is not at all changed by
St3. Vn. OF NATURAL PBlLOSOt»Ht. 107
augmenting or diminijhing that length ; and is therefore the
fame with the refiftance of a circle y defcribed with the fame
diameter, and moving forwards with the fame velocity in the
direSionofa right line perpendicular to its plane.
For the fides are not at all oppofed to the motion ; and' a
cylinder becomes a circle when its length is diminifhed in in^
Jimtum.
PROPOSITION XXXVII. THEOREM XXIX.
If a cylinder move tmiformly forwards in a comprejfed, infinite,
and non^laftic fluid, in the direction of its length, the rc-
fiftanct arifing from the magnitude of its tranfverfe feSiion
is to the force by which its whole motion may be dejiroyed or
generated, in the time that it moves four times its length,
as the denjity of the medium to the denfity of the cyhnder,
nearly.
For let the veffel ABDC (PI. 7, Fig. 5) touch the furfacfe
of ftagnant water with its bottom CD^ and let the water run
oat of this vefTel into the ftagnant water through the cylindric
canal EFTS perpendicular to the horizon ; and let the little
circle PQ be placed parallel to the horizon any where in the
middle of Uie canal ; and produce CA to K^ fo that AK may
be to CK in the duplicate of the ratio, which the excefs of
the orifice of the canal EF above the little circle PQ bears
to the circle AB. Then it is manifeft (by cafe 5, cafe 6, and
cor. J, prop. 36) that the velocity of the water paffing through
the annular fpace between the little circle and the fides of the
yeflel will be the very fame which the wata: would acquire
by falling, and in its fall defcribing the altitude KC or IG.
And (by con 10, prop. 36) if the breadth of the veflel be in-
finite, fo that the lineolse HI may vanifli, and the altitudes
16, HG become equal ; the force of the water that flows
down and prefles upon the circle will be to the weight of a
cylinder whofe bafe is that little circle, and the altitude 4.IG,
as EF* to EF* — ^PQ% very nearly. For the force of the
water flowing downwards uniformly through the whole canal
will be the fame upon the UtUe circle PQ in whatfoever part
of the canal it be placed.
.106 MATflEMATICAL PBTNClTlES Book it.
Let now tbe orifices of the canal £F^ ST be dofed^ and* let
the little circle afcend in tbe flirid oomprefled on eveij iide,
and by its afcent let it oblige the watet that lies above it to
defcend throagh tlie annular f|>ace between the little dfcle
and the fides of the eanal. Then will the velocity ef the
afcending little oirde be to the velocity of the defcemKti^
water as the difference of the circles EF and PQ is to the cir-
cle PQ; and tbe velocity of the afcending little circle will be
to the fum of the velocities^ that is^ to the relative velocity
of the defceoding Water with which it paiTcs by the IHtle cir*
cle in its afcent> as tbe difiere&ce of the circles £F and PQ
10 the circle EF, or as EF* — PQ* to EF*. Let that relaUve
velocity be equal to the velocity with which it was fliewn above
that tbe water would pafs through the annular fpace^ if the
circle were to remain unmoved, that is, to the velocity winch
the water would acquire by fallings and in its fall defcrilring
the altitude 16 ; and the force of the water upon the afcend-
ing circle will be the fame as before (by cor. 5, of tbe laws of
motion) ; that is, the refifl;ance of the afcending little-circle will
be to the weight of a cylinder of water whofe bafe is that little
circle, and its altitude ilO, as EF* to EF* — JPQ», nearly.
But tbe velocity of the little circle will be to the velocity which
the water acquires by falling, and in its fall defcribing tbe sd-
titude IG, as EF* — PQ* to EF*.
Let the breadth of the canal be increafed m infinitum; and
the ratios between EP — PQ* and EF*, and between EI*
and EF* — iPQ*, will become at laft ratios of equality. And
therefore the velocity of the little circle will now be the fame
which the water would acquire in falling, and in its fall de-
fcribing tbe altitude IG; and the refiftanoe will become equa!
to tlie weight of a cylinder whofe bafe is that little eircle, and
its altitude half the altitude IG, from which the cylinder mdft
fall to acquire the velocity of the afcending circle ; and with
this velocity the cylinder in the time of its fall will deTcribe
four times its length. But the refiftance of the cylinder mov-
ing forwards with this velocity in the diredliott of its Icfngth
is the fame with the refiftance of the little circle (by leni. 4)^
and is therefore nearly equal to the force by which its
S€&^ Vn. OV NATURAL PRILOSO^HT, 109
motion may be generated while it de£cribes four times its
length.
If the length of the cylinder be augmented or diminiihed.
Us motioa> and the time in which it defcribes four times its
lengthy will be augmented or diminiihed in the fame ratio •
and theiefbrc the force by which the motion^ fo increafed or
diminiihed^ may be deftroyed or generated^ will continue the
fame ;'becaBfe the time is increafed or diminiihed in the fame
proportion ; and therefore that force remains fiill equal to the
xefiftanoe of the cylinder, becaufe (by lem. 4) that reiiftance
will alfo remain the fame.
If ibe denfity of the cylinder be augmented or diminiihed^
its motioHj and the force by which its motion may be gene<«
rated or deftroyed in the fame time, will be augmented or di-^
miniihed in the fame ratio. Therefore the reiiftance of any
cylinder whatfoever will be to the force by which its whole
motion may be generated or deilroyed, in the time during
which it moves four times its length, as the deniity of the me-
dium to the denfity of the cylinder, nearly. Q.E.D.
A fluid mufl be comprefTed to become continued ; it muii
be continued and non-elaftic, that all the prefiure arifing front
its compreffion may be propagated in an inflant ; and fo, adling
equally upon all parts of the body moved, may produce no
change of the refiftance. The prefTure arifing from the mo-
tion of the body is fpent in generating a motion in the parts of
the fluid, and this creates the refiflanoe. But the prefTure
arifing horn the compreiHon of the fluid, be it ever la
forcible, if it be propagated in an inflant, generates no mo*
tion in the parts of a continued fluid, produces no change at
all of motion therein ; and therefore neither augments nor
leflens the refiflance. This is certain, that the aAion of the
fluid arifing from the compreflion cannot be flro^ogcr on the
hinder parts of the body moved than on its fore parts, and
theiefbro cannot lefTen the refiftance defcribed in this propo*
fition. And if its propagation be infinitely fwifler than the
moticm of the body preffed, it will not be ftronger on the foro
parts than on the hinder parts. But that adlion will be infi-
nitely fwifter, and propagated in an inflant^ if the fluid be coo^
tinned and non-elaftic.
110 MATHEMATICAL PUIKCIPLES Booifc H
Cor. 1. The refiftances m'ade to cylinders going unifbrmlj
forwards in the dire6lion of their lengths through continaed
infinite mediums, are in a ratio compounded of the duplicate
ratio of the velocities and the duplicate ratio of the diameters^
and the ratio of the denfity of the mediums.
Coiu 2. If the breadth of the canal be not infinitely ia-.
creafed, but the cylinder go forwards in the direction of itt
, length through an included quiefcent medium, its axis ftU the
while coinciding with the axis of the canal, its refiftanoe will
be to the force by which its whole motion, in the time in which
it defcribes four times its length, may be generated or de-
ftroyed, in a ratio compounded of the ratio of EP to EF* —
iPQ* once, and the ratio of EP to EF* — PQ* twice, and
the ratio of the denfity of the medium to the denfity of the
. cylinder.
Cor. 3. The fame things fuppofed, and that a length L h
to the quadruple of the length of the cylinder in a ratio com*
pounded of the ratio EF* — iPQ* to EP once, and the ratio
iof EP — PQ* to EP twice ; the refiftance of the cylinder
will be to the force by which its whole motion, in the time
during which it defcribes the length L, may be deftroyed or
generated, as the denfity of the medium to the denfity of the
cylinder. '
SCHOLIUM.
In this propofition we have invefl;igated that refiftance alone
which arifes from the magnitude of the tranfverfe feAion of
the cylinder, negledling that part of the fame which may arife
from tlie obliquity of the motions. For as, in cafe.l, of prop.
36, the obliquity of the motions with which the parts of the
water in the veflel converged on every fide to the hole EF
hindered the efflux of the water through the hole, fo, in this
propofition, the obliquity of the motions, with which the parts
of the water, prefled by the antecedent extremity of the cy-.
linder, yield to the prefiure, and diverge on all fides, jretard^
their pafiage through the places that lie round that ante?.
cedent extremity, towards the hinder parts of the cylinder^
and caufes the fluid to be moved to a greater difi^ance ; which
inci'eafes the refifl^ance, and that in the fame ratio aln^qft \j^
SiS.YlI. OF NATURAL PHILOSOPHY. Ill
which it diminiihed the efflux of the water out of the veifel^
that is^ in the duplicate ratio of 25 to 21, nearly. And as, in
cafe 1, of that proportion, we made the parts of the water
pafs through the hole EF perpendicularly and in the greateft
plenty, by fuppofing all the water in the vefTel lying round
the cataraA to be frozen, and that part of the water whofe
motion was oblique and ufelefs to remain without motion, fa
in this propofition, that the obliquity of the motions may be
taken away, and the parts of the water may give the freeft
paflage to the cylinder, by yielding to it with the mofi di-
ledl and quick motion pofTible, fo that only fo much refiil-
ance may remain as arifes from the magnitude of the tranf*
Terfe fedtion, and which is incapable of diminution, unlefs by
^ONniihing the diameter of the cylinder ; we muft conceive
thofe parts of the fluid whofe motions are oblique and ufelefs,
and produce refliiance, to be at reft among themfelves at both
extremities of the cylinder, and there to cohere, and be joine<}
to the cylinder. Let ABCD (PI. 7, Fig. 6) be a redangle,
und let AE and BE be two parabolic arcs, defcribed with the
axis AB, and with a latus redum that is to the fpace H6
which muft be defcribed by the cylinder in falling, in order to
acquire the velocity with which it moves^ as HG to ^AB»
Let CF and DF be two other parabolic arcs defcribed
trith the axis CD, and a latus redlum quadruple of the former;
^d by the convolution of the figure about the axis EF let
there be generated a folid, whofe middle part ABDC is the
oylindjer we are here fpeaking of, and whofe extreme parts
ABE and CDF contain the parts of the fluid at reft among
themfelves, and concreted into two hard bodies, adhering to
the cylinder at each end like a head and tail. Then if this
folid EACFDB move in the diredlioq of the length of its axis
FE towards the parts beyond E, the refifl^nce will be the
fame which we have here determined in this prppofition^
nearly ; that is, it will have the fame ratio to the force with
which the whole motion of the cylinder may be deflroyed or
generated, in the time that it is defcribing the length 4AC
with that motion uniformly cpntinued, ^s the denfity of the
11(2 MATHBUATICAL PKIMCIPLES Book tl.
fluid has to the denfity of the cylinder, nearly. And (by cor.
7, prop. 36) the refiftance muft be to this force in the ratio of
£ to 3; at the leail,
LEMMA V.
If a cylinder, a fpbere, a fid a fpheroid, of equal breadtht h$
placed fuccejjivefy in tht middle of a cylindrie canal, fo that
• tkeir axe$ may coincide with the axii of the canal, tkefk
bodies will equally hinder the pajfage of the water through
ihe canal.
For the fpaces lying between the fides of the eanalj aa^
the cylinder, fpbere, and fplieroid, through which the watef
pafles, are equal ; and the water will pafs equally thnnig|li
equal fpaces.
This is true, upon the fuppofition that all the water above
the cylinder, fphere, or fpheroid, whofe fluidity is not neoeH
fary to make the paiTage of the water the quickeft poflibfey
is congealed, as was explained above in cQr. 7> prop. 38.
LEMMA VL
The fame fuppofition remaining, the forementioned bodies arg
tqually aSed on by the waterflowing through the canal.
This appears by lem. 5, and die third law. For the tiFater
and the bodies a& upon each other mutually and equally.
LEMMA Vn.
If the water he at reft in the canal , and thefe bodies mjove with
equal velocity and the contrary way through the canal, their
refiftanees will be equal among themfelves.
lliis appears from the laft lemma, for the relative motioQg
remain the fame among themfelves.
SCHOLIUM.
The cafe is the fame of all convex and round bodies,
wbofe axes coincide with the axis of the canal. Somediffinr*
«nee may arife frcmi a greater or lels friction ; but in thefe
iemtnata we fuppofe the bodies to be perfectly fmooth, and
the medium to be void of all tenacity and fri&ion ; and that
thofe parts of the fluid whicb by their oblique and (uperfluotts
motions may difturb, hinder, and retard the flax of the wate*
through the canal, are at r^ft amongfl; themfelves; being
fixed like water by froft, and adhering to the fore and hinder
.V.
piafy rir. m.j[.
^la^gi.ffB.
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Qi
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Se8, VIL OP NATURAL PHILOSOPHY, 113
parts of the bodies in the manner explained in the fcholium
of the laft propofition; for in what follows we confider
the very leaft refiftance that round bodies defcribed with the
greateil given tranfverfe feftions can poffibly meet with.
Bodies fwimming upon fluids, when they move flraight for-
wards^ caufe the fluid to afcend at their fore parts and fubflde
at their hinder parts, efpecially if they are of an obtufe figure;
and thence they meet with a little more refiftance than if they
were acute at the head and tail. And bodies moving in elaftic
fluids, if they are obtufe behind and before, condenfe the
fluid a little more at their fore parts, and relax the fame at
their hinder parts; and therefore meet alfo with a little more
lefiftailce than if they were acute at the head and tail. But ,
in thefe lemma<$ and propofitions we are not treating of
elaftic but non-elaftic fluids; not of bodies floating on the
furface of the fluid, but deeply immerfed therein. And when
the refifliance of bodies in non-elaftic fluids is once known, we
may then augment this refiftance a little in elaftic fluids, as
our air; and in the furfaces of ftagnating fluids, as lakes and
feas.
PROPOSITION XXXVIII. THEOREM XXX.
If a globe move uniformly forward in a compreffed, irifimte,
and non-^laftic fluid, its refiftance is to the force by which
its whole motion may be dejlroyed or generated, in the time
that it defcriben eight thirdparts of its diameter, as the denji*
ty of the fluid to the denfity'of the globe, very nearly.
For the globe is to its circumfcribed cylinder as two to
three ; and therefore the force which can deftroy all the mo-
tion of the cyUnder while the fame cylinder is defcribing the
length of four of its diameters, will deftroy all the motion of
the globe while the globe is defcribing two thirds of this
lengtli, that is, eight third parts of its own diameter. Now
the refiftance of the cylinder is to this force very nearly as the
denfity of the fluid to the denfity of the cylinder or globe (by
prop. 37)^ and the refiftance of the globe is equal to the refift-
ance of the cyhnder (by lem. 5, 6, and 7). Q.E.D.
Cob. 1. The refifl;ances of globes in infinite comprefled me-
diums are in a ratio compounded of the duplicate ratio of the
Vol. II. I
114 MATHEMATICAL PBIICCIPLSt Book Up
velocity^ and the duplicate ratio of the diameter^ and the ratio
of the denfity of the mediums.
Cor. Q. The greateil velocity with which a globe can de-
fcend by its comparative weight through a refifting flnid^ i«
the* fame which it may acquire by falling with the fame
weighty and without any refiftance, and in its fall defcribiog
a fpace that is to four third parts of its diameter as the denfi-
ty of the globe to the denfity of the fluid. For the globe ia
the time of iti^ fall^ moving with the velocity acquired in fall*
ing^ will defcribe a fpace that will be to eight third parts of its
diameter as the denfity of the globe to the denfity of the floid;
and the force of its weight which generates this motion will
be to the force that can generate the fame motion, in the tfaac
that the globe defcribes eight third parts of its diameter, with
the fame velocity as the denfity of the fluid to the denfity of
the globe ; and therefore (by this propofition) the force of
weight will be equal to the force of refifiance, and therefbie
cannot accelerate the globe.
Cob. 3. If there be given both the denfity of the globe
and its velocity at the beginning of the motion, and the den-^
fity of the comprefled quiefcent fluid in which the ^be
moves, there is given at any time both the velocity of the-
globe and its refiflance, and the fpace defcribed by it (by cor.
7, prop. 35).
Cob. 4. A globe moving in a comprefled quiefcent floid of
the fame denfity with itfelf will lofe half its motion before
it can defcribe the length of two of its diameters (by the fane
cor. 7).
PROPOSITION XXXIX. THEOREM XXXI.
If a globe move uniformly forward through a fluid mchfei
and compreffed in a cylindric canal, its refiftance u to ilk
force by which its whole motion may be generated or dtfiffi^
id, in the time in which it defcribes eight third parts rfiU
diameter, in a ratio compounded of the ratio of the ori/le$
of the canal to the excefs of that orifice above half tlU-
greateji circle of the globe ; and the duplicate ratio of ike
orifice of the canal to the excefs of that orplce above ike
SeS. VII. OF NATURAL PHILOSOPHY. Il5
grcatefi circle of the globe; and the ratio of the denfity of
the fluid to the denfity of the globe y nearly.
Tliis appears by cor. 2^ prop. ST, and the demonftratioa.
proceeds in the fame manner as in the foregoing propofition.
SCHOLIUM.
In the two lafi; propofitions we fuppofe (as was done before
in lem. 5) that all the water which precedes the globe^ and
whole fluidity increafes the redftance of the fame, is congeal-
ed. Now if that water becomes fluid, it will fomewhat in-
creaie die refiftance. But in thefe proportions that increafe
is fo fmall, that it may be neglected, becaufe the convex
faperficies of the globe produces the very fame effed); aimed as
ihe congelation of the water.
PROPOSITION XL. PROBLEM IX.
To find by phenomena the refijiance of a globe moving through
a perfeSly fluid compreffed medium.
Let A be the weight of the globe in vacuo, B its weight in
the lefifting medium, D the diameter of the globe, F a fpace
wfaidi 18 to ^D as the denfity of the globe to the denfity of
the koediun, that is, as A to A — B, 6 the time in which
die globe falling with the weight B without refiflance de-
Jciibes the fpace F, and H the velocity which the body ac-
^pirei by that fall. Then H will be the greateft velocity with
which the globe can poflibly defcend with the weight B in the
lefiffing medium, by cor. 2, prop. 38 ; and the refiflance which
llie globe meets with, when defcending with that velocity,
nin be equal to its weight B; and the refiftance it meets
with in any other velocity will be to the weight B in the du-
^icate ratio of that velocity to the greatefl velocity H, by cor.
if prop. 38.
is the refiflance that arifes from the ina6livity of the
of the fluid. That refiflance which arifes from the
|daftici^j tenacity, and friAion of its parts, may be thus in-
;feftig&ted.
Let the globe be let fall fo that it may defcend in the fluid
by the weight B ; and let P be the time of falling, and let
that time be exprefTed in. feconds, if the time 6 be given in
lleconds. Find the abfolute number N agreeing to the lo-
I 2
Il6 MATHEMATICAL PUINCIPLKS Book Ih
op
garitliin 0^13429448 1 9 yr 9 and let L be the logarithm of
N + 1 .
the number — rj — 5 and the velocity acquired in falling will
N — 1 ttPF
be ^ H, and the height defcribcd will be -ts;
1^86294361 IF + 4,6051 701 86LF. If the fluid be of a Ef-
ficient depth, we may neglcd the term 4,6031 70 186LF ; and
2PF
-^ 1,386294361 iF will be the altitude dcfcribed, nearly.
Thefe things appear by prop. 9, book 2, and its corollaries,
and are true upon this fuppofition, that the globe meets with
no other refiftancc but that which arifes from the inactivity
of matter. Now if it really meet with any refiftancc of ano-
ther kind, the defcent will be flower, and from the quantity
of that retardation will be known the quantity of this new re-
fiftancc.
That the velocity, and defcent of a body falling in a fluid
might more eafily be known, I have compofed the following
table ; the firft. column of which denotes the times of defcent;
the fecond fliews the velocities acquired in falling, the greateft
velocity being 100000000 ; the third exhibits the fpaces dc-
fcribed by falling in thofe times, 2F being the fpace which
the body defcribes in the time G with the greateft velocity ;
and the fourth gives the fpaces defcribed with the greateft ve-
locity in the fame times. The numbers in the fourth coluflui
are j^j andbyfubdudingthe number 1,3862944—4,605 1702li|
are found the numbers in the third column; and thefe num«
bers muft be multiplied by the fpace F to obtain the fpaces
defcribed in falUng, A fifth column is added to all tbefe>
containing the fpsces defcribed in the fame times by a body
falling in vacuo with the force of B its comparative weight.
Sea. VII
OF NATURAL PHILOSOPHY.
117
The fpaces
The Timei
p
Velocitiet of the
body falling in
The fpaces defcrib-
ed in falling in
defcribed
with the
The fpaces de-
fcribed by fall-
X ■
the fluid.
the fluid.
g^reateft
motion.
ing in vacuo.
0,00 IG
999991*
0,00000 IF
0,002F
0,00000 iF
0,01 G
999967
0,000 IF
0,02F
0,000lF
0,iG
9966799
0,0099834F
0,2F
0,QlF
0,2G
19737532
0,039736 IF
0,4F
0,04F
0,3G
29131261
0,08868 I5F
0,6F
0,09F
0,4G
37994896
0,I559070F
0,8F
0,16F
0,5G
46211716
0,2402290F
1,0F
0,25F
0,6G
53704957
0,3402706F
1,2F
0,36F
0,7G
60436778
0,4545405 F
1,4F
0,49F
0,8G
66403677
0,58 1507 IF
1,6F
0,64F
0,9G
71629787
0,7196609F
1,8F
0,81 F
IG
76159416
0,867561 7F
2F
IF
2G
96402758
2,6500055F
4F
4F
3G
99505475
4,6l86570F
GF
9F
4G
99932930
6,6l43765F
8F
16F
.5G
99990920
8,6l37964F
lOF
25F
6CJ
99998771
10,61 37 179F
I2F
36F
7G
99999834
12,6l37073F
14F
49F
86
99999980
14,6lS7059F
liiF
64F
9G
99999997
16,61 37057 F
18F
81F
lOG
999999991
18,61 37O06F 20F
lOOF
SCHOLIUxM.
In order to inveftigate the refiftances of fluids from expe-
riments, I procured a fquare wooden vefT^?!, whofe length and
breadth on the in{i«Ic was 9 inches Efigli/Ji meafure, and its
depth 9 feet i ; lliis 1 filled with rain-water : and having pro-
vided globes made iij) of wax, and lead included therein, I
noted the times of the defcents of thefe globes, the height
through which they defccnded being 112 inches. A folid
cubic foot of Engli/h nieafurc contains 76 pounds troi/ weight
pf rain-water; and a foHd inch contains ^ ounces troy weight,
QF 2a3V graiqs ; and a globe of water of one inch in diameter
contains 132,645 grains in air, or 132,8 grains in vacuo ; and
any other globe will be as the cxcefs of it^ weight in vacuo
above its weight in water.
ExPER. 1. A globe whofe weight was 156i: grains in air,
and 77 grains in water, defcribed the whole height of 112
I 3
118 MATHEMATICAL PBXNCTPLES Book II.
inches in 4 fcconds. And, upon repeating the experiment,
the globe fpent again the very fame time of 4 feconds la
falling.
The weight of this globe in vacuo u 156K- grains ; and ex-
cefs of this weight above the weight of the globe in water is
79ii grains. Hence the diameter of the globe appears to be
0,84224 parts of an inch. Then it will be, as that ezcefs to
the weight of the globe in vacuo, fo is the denfity of the
water to the denfity of the globe ; and fo is } parts of the di-
ameter of the globe (viz. 2,24597 inches) to the fpace 2F,
which will be therefore 4,4256 inches. Now a globe fialliDg
in vacuo with its whole weight of 156^ grains in one feoond
of time will defcribe 1931 inches ; and falling in water in the
fame time with the weight of 77 grains without refifiancCj
will defcribe 95,219 inches ;- and in the time 6, which is to
one fecond of time in the fubdilplicate ratio of the fpaoe F,
or of 2,2128 inches to 95,219 inches, will defcribe 2^2 128
inches, and will acquire the greateft velocity H with which it
is capable of defcending in water. Therefore the time 6 is
0^^15244. And in this time 6, with that greateft velocity H,
the globe will defcribe the fpace 2F, which is 4,4256 inches ;
and therefore in 4 feconds will defcribe a fpace of 1 16^1245
inches. Subdudl the fpace ],d862944F, or 3,0676 inches^ and
there will remain a fpace of 1 13,0569 inches, which the globe
falling through water in a very wide veflel will defcribe in 4
feconds. But this fpace, by reafon of the narrownefs of the
wooden veflel before-mentioned, ought to be diminilhed in a
ratio compounded of the fubduplicate ratio of the orifice of
the veflel to the excefs of this orifice above half a great circle
of the globe, and of the fimple ratio of the fame orifice to its
excefs above a great circle of the globe, that is, in a ratio of
1 to 0,9914. This done, we have a fpace of 112,08 inches,
which a globe falling through the water in this wooden veffet
in 4 feconds of time ought nearly to defcribe by this theory;
but it defcribed 112 inches by the experiment.
ExpER. 2. Three equal globes, whofe weights were feve»
rally 76J grains in air, and 5iV grains in water, were let &11
ScB. VIL OF NATURAL PHILOSOPHY. 119
fucceffivelj; and every one fell through the water in 15 fe-
conds of timej defcribing in its fall a height of 1 12 inches.
By computation^ the weight of each globe in vacuo is 7&A
grains ; the excefs of this weight above the weight in water
is 7 1 grains H ; the diameter of the globe 0,8 1 S96 of an inch ;
f parts of this diameter 2,16789 inches; the fpace 2F is
Sjd217 inches ; the fpace which a globe of 5tV grains in weight
urould defcribe in one fecond without refiftance, 12,808 inches^
and the time G0",301056. Therefore the globe, with the
greateft velocity it is capable of receiving from a weight of
5iV grains in its defcent through water, will defcribe in the
time 0"^1(>56 the fpace of 2,3217 inches; and in 15 fe-
conds the fpace 115,678inches. SubduA the fpace l,3862g44F,
or IfiOQ inches^ and there remains the fpace 1 14,069 inches ;
which therefore the falling globe ought to defcribe in the fame
time^ if the veffel were very wide. But becaufe our veffel was
oarroWj the fpace ought to be diminiflied by about 0,895 of
an inch. And fo the fpace will remain 1 13,174 inches, which
A globe falling in this veffel ought nearly to defcribe in 15
feconds, by the theory. But by the experiment it defcribed
1 le inches. The difference is not fenfible.
ExPER. 3. Three equal globes, whofe weights were feve-
rally 121 grains in air, and 1 grain in water, were fuccefiively
let fall ; and they fell through the water in the times 46'^ 47'',
and 50^^ defcribing a height of 1 12 inches.
By the theory, thefe globes ought to have fallen in about
40". Now whether their falling more flowly were occafioned
from henoe^ that in flow motions the refiftance arifing from
the force of inadtivity does really bear a lefs proportiovto the
refiflance arifing from other caufes ; or whether it is to be at-
tributed to little bubbles that might chance to ftick to the
globes, or to the rarefadlion of the wax by the warmth of
the weather, or of the hand that let them fall ; or, lafUy^
nrhether it proceeded from fome infenfible errors in weighing
the globes in the water, I am not certain. Therefore the
ureight of the globe in water fhould be of feveral grains, that
the experiment may be certain, and to be depended on.
I 4
120 MATHEMATICAL PRINCIPLES Book II.
ExPKR. 4. I began the fc/rcgoing experiments to inveili-
gate the refirtauces of fluids^ before 1 was acquainted with
the theory laid down in ihc propofitions immediately preced-
ing. Aftenvards^ in order to examine the theory after it was
difcovcrcd, I procured a wooden veffel, whofe breadth on the
infide was 87 inches, and its depth 15 feet and ^. Then I
made four globes of wax, with lead included, each of which
weighed I39i grains in air, and 7t grains in water. Thefe I
let fall, mcafuring the times of their falling in the water with
a pendulum ofcillating to half feconds. The globes were coldj
and had remained fo fome time, both when they were weighed
and when they were let fall ; becaufe warmth rarefies the
wax, and by rarefying it diminiflies the weight of the globe
in the water ; and wax, when rarefied, is not infiantly re-
duced by cold to its former denfity. Before they were let fall,
they were totally immerfed under water, left, by the weight
of any part of them that might chance to be above the water,
their defcent fhould be accelerated in its beginning. Then,
when after their immerfion they were perfedlly at reft, they
were let go with the greateft care, that they might not receive
any impulfe from the hand that let them down. And they
fell iucceffively in the times of 47^^, 48i, 50, and 51 ofcilla-
lations, defcribing a height of 15 feet and 2 inches. But the
weather was now a little colder than when the globes were
weighed, and therefore I repeated the experiment another
day ; and then the globes fell in the times of 49, 49^, 50, and
53 ; and at a third trial in the times of 49t> 50, 51, and 53
ofcillations. And by making the experiment feveral times
over, I found that the globes fell moftly in the times of 49J-
and 50 ofcillations. When they fell flower, I fufpedl them
to have been retarded by ftriking againft the fides of the
veflTel.
Now, computing from the theory, the weight of the globe
in vacuo is 139|- grains ; the excels of this weight above the
weight of the globe in water IS^iS- grains ; the diameter of the
globe 0,99363 of an inch ; J parts of the diameter 2,66315
inches ; the fpace 2F 2,8066 inches; the fpace which a globe
weighing 71* grains falling withput refiilance defcribes in a fe-
S<:£t, VIL OF NATURAL PHILOSOPHY. 121
cond of time 9,88164 inches; and the time Go",376843.
Therefore the globe with the greateft velocity with which it
is capable of defcending through the water by the force of a
weight of Tj-grains^ will in the time 0",376843defcribe a fpace
of 2,8056 inches, and in one fecond of time a fpace of 7^44766
inches, and in the time 25", or in 50 ofcillations, the fpace
186,1915 inches. Subdud the fpace 1,386294F, or 1,9454
inches, and there will remain the fpace 184,2461 inches
vhich the globe will defcribe in that time in a very wide vef-
fel. Becaufe our veflel was narrow, let this fpace be diAi-
niflied in a ratio compounded of the fubduplicate ratio of the
orifice of the veflel to the excefs of this orifice above half a
great circle of the globe, and of the fimple ratio of the fame
orifice to its excefs above a great circle of the globe ; and we
fliall have the fpace of 181,86 inches, which the globe ought
by the theory to defcribe in this veflel in the time of 50 ofcil-
lations, nearly. But it defcribed the fpace of 182 inches, by
experiment^ in 49f or 50 ofcillations.
ExPER. 5. Four globes weighing 154| grains in air, and
21f grains in water, being let fall feveral times, fell in the
timeisof 28|, 29, 29 1, and 30, and fometimes of 31, 32, and
33 ofcillations, defcribing a height of 15 feet and 2 inches.
They ought by the theory to have fallen in the time of 29
ofcillations, nearly.
ExPKR. 6. Five globes, weighing 212|- grains in air, and
79i i» water, being feveral times let fall, fell in the times of
15, 15|, l6, 17, and 18 ofcillations, defcribing a height of 15
feet and 2 inches.
By the theory they ought to have fallen in the time of 15
ofcillations, nearly.
ExPER. 7. Four globes, weighing 293|- grains in air, and
35i grains in water, being let fall feveral times, fell in the times
of 29i,30, 30f, 31, 32, and 33 ofcillations, defcribing a height
of 15 feet and 1 inch and -|.
By the theory they ought to have fallen in the time of 28
ofcillations, nearly.
In fearching for the caufe that occafioned thefe globes
of the fame weight and magnitude to fall, fome fwifter and
1V2'2 MATHEMATICAL PRINCIPLES JSooA IL
fome flower^ I bit upon this ; that the globes, when they were
firft let go and began to fall^ ofcillated about their centres ;
that iide which chanced to be the heavier defcending firft^
aikd producing an ofcillating motion. Now by ofcillating
thus, the globe communicates a greater motion to the water
than if it defcended without any ofcillatious; and by this
communication loi'es part of its own motion with which it
fliould defcend ; and therefore as this ofcillation is greater or
lefs> it will be more or lefs retarded. Befides, the globe always
recedes from that fide of itfelf which is defcending in the of
cillation, and by fo receding comes nearer to the fides of the
velTel, fo as even to fixike againft them fometimes. And the
heavier the globes are, the ftronger tliis ofcillation is ; and
the greater they are, the more is the water agitated by it.
Therefore to diminifh this ofcillation of the globes, I made
new ones of lead and wax, flicking the lead in one fide of the
globe very near its furface ; and I let fall the globe in fuch
a manner, that, as near as poffible, the heavier fide might be
loweft at the beginning of the defcent. By this means the
ofcillations became much lefs than before, and the times in
which the globes fell were not fo unequal : as in the follow-
ing experiments.
£xP£R. 8. Four globes weighing 139 grains in air, and 6|
in water, were let fall feveral times, and fell moftly in the
time of 5 1 ofcillations, never in more than 5£, or in fewer
than 50, defcribing a height of 182 inches.
By tlie theory they ought to fall in about the time of 52 of-<
cillations.
ExPER. 9. Four globes weighing 27'3i grains in air, and
140| in water, being feveral times let fall, fell in never fewer
than 12, and never more than 13 ofcillations, defcribing a
height of 182 inches.
Thefe globes by the theory ought to have fallen in .the Um^
of 1 1-|- ofcillations, nearly.
Expeh. 10. Four globes, weighing 384 grains in air, and
119} in water, being let fall feveral times, fell in the times of
17-}^ 18, 18}, and 19 ofcillations, defcribing a height of 181 1
inches. And when they fell in the time of 19 ofciUationa^ I
SeS.YIh OF NATURAL PHILOSOPHY. 193
fometimes heard tbem hit againft the fides of the veflel be-
fore they reached the bottom.
By the theory they ought to have fallen in the time of
151 ofcillations^ nearly.
ExPER. 11. Three equal globes^ weighing 48 grains in the
iiir, and S|4 in water, being feveral times let fall, fell in the
times of 43i, 44, 44|, 45, and 46 ofcillations, and moflly ia
44 and 55, defcribing a height of 1 82| inches, nearly.
By the theory they ought to have fallen in the time of 46
pfcillations and ^, nearly.
ExPER. 1*2. Three equal globes, weighing 141 grains in
lur^ and 4j- in water, being let fall feveral times, fell in the
times of 6l, 62, 63, 64, and 65 ofcillations, defcribing a fpace
of 182 inches.
And by the theory they ought to have fallen in 64i ofcilla-
tions, nearly.
From thefe experiments it is manifeft, that when the globes
fell flowly, as in the fecond, fourth, fifth, eighth, eleventh,
and twelfth experiments, the times of falling are rightly ex*
hibited by the theory ; but when the globes fell more fwiftly,
as in the fixth, ninth, and tenth experiments, the refiftance
was fomewhat greater than 'in the duplicate ratio of the ve-
locity. For the globes in falling ofcillate a little ; and this
ofciilation, in thofe globes that are light and fall fiowly, fooQ
. ceafes by the weaknefs of the motion ; but in greater and
heavier globes, the motion being ftrong, it continues longerj
and is not to be checked by the ambient water till after feve«
ral ofcillations. Befides, the more fwiftly the globes move,
the lelk are they prefied by the fluid at tlieir hinder parts ;
and if the velocity be perpetually increafed, ihey will at laft
leave an empty fpace behind them, unlefs the comprefiicTn of
the fluid be increafed at the fame time. For the compreflioa
of the fluid onght to be increafed (by prop. 32 and 33) in the
duplicate ratio of the velocity, in order to preferve the refifi-.
ance in the fame duplicate ratio. But becaufe this is
not done, the globes that move fwiftly are not fo much preiC*
ed at their hipder parts as the others ; and by (he dcfedl: of
]<24 MATHEMATICAL PRINCIPLES Book II.
this prefTiire it comes to pafs that their refiftance is a little
greater than in a duplicate ratio of their velocity.
h>o that t)ic theory agrees with the pha?nomcna of bodies
falling in water. It remains that we examine the phaenomena
of hodics falling in air.
Ex PER. 13. From the top of St. Pflj/fs Church in Lon-
don, in June 1710, there were let fall together two glafs
globes, one full of quickfilver, the other of air ; and in their
fall they defcribed a height of 220 Engli/h feet. A wooden
table was fufpendcd upon iron hinges on one fide, and the
other fide of the fame was fupported by a wooden pin.
The two globes lying upon this table were let fall together by
pulling out the pin by means of an iron wire reaching from
thence quite down to the ground; fo that, the pin being re-
moved, the table, which had then no fupport bat the iron
hinges, fell downwards, and, turning round upon the hinges,
gave leave to the globes to drop off from it. At the £une
inftant, with the fame pull of the iron wire that took out the
pin, a pendulum ofcillating to feconds was let go, and began
to ofcillate. The diameters and weights of the globes, and
their times of falling, are exhibited in the following table.
The gtobts filicd with mercury. \
The globes full of air. |
Weights.
Diameters.
Times in
falling.
Weights. .
Diameters.
Timet in
falling.
908 grains
0,8 of an inch
4//
5 10 grains
5,1 inches
8"i
983
0,8
4 —
642
5,2
8
8f)6
0,8
4
599
5,1
8
747
0,75
4 +
5,0
8{
808
0,75
4
483
5,0
H
784
0,75
4 +
641
1
5,2
8
But the times obferved muft be corrected ; for the globes of
mercury (by Galileo's theory), in 4 feconds of time, will de-
fcribe 257 J'^ngliJIi feet, and 220 feet in only 3"42'". So that the
wooden table, when the pin was taken out, did not turn upon
its hinges fo quickly as it ought to have done ; and the flow-
nels of that revolution hindered the defcent of the globes at
the beginning. For the globes lay about the middle of tjie
table, and indeed were rather nearer to the axis upon which
it turned than to the pin. And hence the times of falling
SeS. VII. OP NATURAL PHILOSOPHY. 125
were prolonged about 18'"; and therefore ought to be correft-
ed by fubdu&ing that excefs^ efpecially in the larger globes^
which^ byreafon ofthelargenefs of their diameters^ lay longer
upon the revolving table than the others. This being done,
the times in which the fix larger globes fell will come forth
8" 12'", 7" 42"', 7" 42"', 7" 57'", 8" 12'", and 7" 42'".
Therefore the fifth jn order among the globes that were
full of air being 5 inches in diameter^ and 483 grains ia
weight, fell in 8'' 12'", defcribing a fpace of 220 feet. The
weight of a bulk of water equal to this globe is 166OO grains ;
and the weight of an equal bulk of air is HI8^ grains, or
ig^ grains; and therefore the weight of the globe in vacuo
is 502-^ grains ; and .this weight is to the weight of a bulk
of air equal to the globe as 502tu to IQrV; and.fo is' 2F to |- of
the diameter of the globe, that is, to 13* inches. Whence 2F.
becomes 28 feet 1 1 inches. A globe falling in vacuo with its
whole weight of 502-iiy grains, will in one fecond of time de-
fcribe ig3-|- inches as above; and with the weight of 483
grains will defcribe 185,905 inches ; and with that weight 483
grains in vacuo will defcribe the fpace F, or 14 feet 5| inches,
in the time of 57'" 58"", and acquire the greatefl: velocity it is
capable of defcending with in the air. With this velocity
the globe in 8" 12'" of time will defcribe 245 feet and 5^
inches. SubduA 1,3863F, or 20 feet and | an inch, and there
remain 225 feet 5 inches. This fpace, therefore, the falling
globe ought by the theory to defcribe in 8" 12'". But by
the experiment it defcribed a fpace of 220 feet. The difier-
ence is infenfible.
By like calculations applied to the other globes full of air,
I compofed the following table.
The ireights|The diame-
oftlie globes.
alOgrains
642
599
515
483
641
ten.
The times of
falling from
i. a height of
220 feet.
5,linches.8" 12'"
5,2
5,1
5
5
5,2
7
7
7
8
42
42
57
12
42
The fpaces which they
would defcribe by
the theory.
226 feet 11 inch.
230
227
224
225
1 230
9
10
5
5
7
The ezceflei.
6feet 11 inch.
10
7
4
5
10
9
10
5
5
126 MATHEMATICAL PRINCIPLES Booklh.
ExPER. 14. Anno 1719, In the mont.h of July, Dr, D^Ja^
gulitrs made ibme experiments of this kind again^ by form*
iog hogs* bladders into fpherieal orbs ; which was done by
means of a concave wooden fphere^ which the bladders, beiag
wetted well firil, were put into. After that, being/blown full
of air, they were obliged to fill up the fpherieal cavity that
contained them ; and then, when dry, were taken out. Thefe
were let fall from the lantern on the top of the cupola of the
fame church, namely, from a height of Q7Q feet ; and at the
fame moment of time there was let fall a leaden globe, whofe
weight was about 9, pounds troi/ weight. And in the mean
time fome perfons (landing in the upper part of the church
where the globes were let fall obferved the whole times of
falling ; and others {landing on the ground obferved the dif-
ferences of the times between the fall of the leaden weight
and the fall of the bladder. The times were meafured by
pendulums ofcillating to half feconds. And one of thofe that
flood upon the ground had a machine vibrating four times in
one fecond ; and another had another machine accurately
made with a pendulum vibrating four times in a fecond alfo.
One of thofe alfo who ftood at the top of the church had a
like machine ; and thefe inftruments were fo contrived, that
their motions could be flopped or renewed at pleafure. Now
the leaden globe fell in about four feconds and ^ of time ; and
from the addition of this time to the difference of time above
fpoken of, was colle6led the whole time in which the bladder
was falling. The times which the five bladders fpent in fall-
ing, after the leaden globe had reached the ground, were, the
firfltime, 141", 121", Uf, 17 f, and l6|"; and the fecond
time, 14i", 14f', 14", 19", and l6|". Add to thefe 4i", the
time in which the leaden globe was falling, and the whole
times in which the five bladders fell were, the firfl time, I9", 17"*
18f', 22", and«]V'; and the /econd time, 18f, 18f, 18^*
(J5|", and 21 ". The times obferved at the top of the church
were, the firfl: time, 19^", 17i", ISf, 22f ', and 2lf' ; and the
fecond time, 1 9", 1 8f' , 1 8|", 24", and 2 ] }". But the bladders did
not always fall diredlly down, but fometimes fluttered a little
in the air, and waved to and fro as they were defcending.
And by thefe motions the times of their falling were prolonged.
SeS. Vn. OF NATURAL PHILOSOPHT. 127
and increafed by half a fecond fometimes^ and fometimcs by
a whole fecond. The fecond and fourth bladder fell moftdi-
reiSIy the firft time^ and the firft and third the fecond time.
The fifth bladder was wrinkled^ and by its wrinkles was a little
retarded. I found their diameters by their circumferences
meafured with a very fine thread wound about them twice. In
the following table I have compared the experiments with the
theory ; making the denfity of air to be to the deniity of rain-
water as 1 to S60, and computing the fpaces which by the
theory the globes ought to defcribe in falling.
The fpaces which by
the tneory ought to
have been deficribed
in thofe times.
The weight!
of theblad-
doiB*
128 grains
156
137l
97f
99*
The diameters.
5^28 inclies
5,3
5,26
5
The times of
falling from
a height of
272 feet.
19"
17
18
22
271ieetll in.
272 Of
272 7
277 4
282
The diffisrence be-
tween the theoir
and the experi-
ments.
— Oft. i in.
+ Of
+ 07
+ 54
+ 10
1
Our theory, therefore, exhibits rightly, within a very little,
all the refiftance that globes moving either in air or in water
meet with; which appears to be proportional to the den-
fities of the fluids in globes of equal velocities and mag-
nitudes.
In the fcholium fubjoined to the fixth fedlion, we (Iiewed,
by experiments of pendulums, that the reiiftances of equal
and equally fwifl globes moving in air, water, and quickfilver,
are as the denfities of the fluids. We here prove the fame
more accurately by experiments of bodies falling in air and
water. For pendulums at each ofcillation excite a motion in
the fluid always contrary to the motion of the pendulum in
its return ; and the refiftance arifing from this motion, as alfo
the refiftance of the thread by which the pendulum is fuf-
pended, makes the whole refiftance of a pendulum greater
than the refiftance deduced from the experiments of falling
bodies. For by the experiments of pendulums defcribed in
that fcholium, a globe of the fame deniity as water in de-^
fcribing the length of its femi-diameter in air would lofe the
part of its motion. But by the theory delivered in thLi
128 .MATHEMATICAL PRINCIPLES Book IL
feventh fedtion^ and confirmed by experiments of falling bo-
dies, the fame globe in defcribing the fame length would lofe
only apart of its motion equal to , fuppofing the denfity
4 5 8 6
of water to be to the denfity of air as 8(J0 to 1. Therefore
the refinances were found greater by tlie experiments of pen*
dulums (for the rcafous juft mentioned) than by the experi-
ments of falling globes ; and that in the ratio of about 4 to 3.
But yet fince the refiilances of pendulums ofcillating in air,
water, and quickillver, are alike increafed by like caufes, the
; proportion of the refifl:ances in thefe mediums will be rightly
enough exhibited by the experiments of pendulums, as well
as by the experiments of falling bodies. And from all this it
. may be concluded, that the refifi:ances of bodies, moving in
any fluids whatfoever, though of the mofl: extreme fluidity^
are, cateris paribus, as the denfities of the fluids.
Thefe things being thus eftabliflied, we may now determine
what part of its motion any globe projedled in any fluid what-
foever would nearly lofe in a given time. Let D be the dia-
meter of the globe, and V its velocity at the beginning of its
motion, and T the time in which a globe with the velocity V
can defcribe in vacuo a fpace that is to the fpace 4D as the
denfity of the globe to the denfity of the fluid; and the globe
projeAed in that fluid will, in any other time t, lofe the part
tV TV . .
rp > the part ^ remaining ; and will defcribe a fpace^
which will be to that defcribed in the fame time in vacuo with
the uniform velocity V, as the logarithm of the number
T + t
— =s — multiplied by the number 2,302585093 is to the number
t
^j by cor. 7 9 prop. 35. In flow motions the refifl;ance may
be a little lefs, becaufe the figure of a globe is more adapted
to motion than the figure of a cylinder defcribed with the
fame diameter. In fwifk motions the refifl;ance may be a little
greater, becaufe the elafi;icity and comprefliion of the fluid do
not increafe in the dupUcate ratio of the velocity. But thefe
little niceties I take no notice of.
Se&.\IL OF K ATT] RAL PHILOSOPHY. 129
And though air^ Water^ quickfilver^' and the like fluids, by
the divifion of their parts in infinitum, (hould be fubtilized^
and become mediums infinitely fluid, ueverthclefs, the refift-
ance they would make to projedled globes would be the fame.
For the refiftance confidered in the preceding pro]>ofltions
arifes from the inadlivity of the matter ; and the ina^vity of
matter is eflential to bodies, and always proportional to the
quantity of matter. By the divifion of the parts of the fluid
the refiftance arifing from the tenacity and fridlion of tlie
parts may be indeed diminiihed ; but the quantity of matter .
will not be at all diminiflied by this divifion ; and if the quan-..
tity of matter be the fame, its force of inadivity will be the
fame ; and therefore the refiftance Irere fpoken of will be the
famCj as being always proportional to that force. To diminifli
this refiftance, the quantity of matter in the fpaces through
which the bodies move mutt be diminiflied : and therefore •
the celeftial fpaces, through which the globes of the planets
tad comets are perpetually pafling towards all parts, with
the utmofi freedom, and without the leatt fenfible diminution
of their motion, muft be utterly void of any corporeal fluids .
excepting, perhaps, fome extremely rare vapours and the rays
of light.
Projectiles excite a motion in fluids as they pafs through
them ; and this motion arifes from the excefs of the preflure.
of the fluid at the fore-parts of the proje<ftile above the pref-
fure of the fame at the hinder parts ; and capnot be lefs in
mediums infinitely fluid than it is in air, water,. and quick*,
filver^ in proportion to the denfity of matter in each. Now;
this excefs of preflure does, in proportion to its quantity, not
only excite a motion in the fluid, but alfo aAs upon the pro-'
jedUle fo as to retaixl its motion ; and therefore the refifi:ance
in every fluid is as the motion excited by the proje6);ile in the
fluid ; and cannot be lefs in the moft fubtile aether in proper*
tion to the denfity of that aether, than it is in air, water,
and qnickfilver, in proportion to the denfities of thofe
fluids.
Vol. ir. K
128 MATHEMATICAL PRINCIPLES Book IL
feventh fcdtion^ and coniiriucd by experiments of falling bo-
dies, the fame globe in dolcribing the fame length wculd lofe
only apart of its motion equal to , fuppofmg the denfity
4 5 8 6
of water to be to the denfity of air as 8()0 to 1. Therefore
the refiftanei's were found greater by the experiments of pen-
dulums (for the reafons jull mentioned) than by the experi-
ments of falling globes ; and that in the ratio of about 4 to 3.
But yet finee the refillances of pendulums ofcillating in air,
water, and (juickiilver, are alike increafed by hke eaufes, the
proportion of the refiftanees in thefe mediums will be rightly
enough exhibited by the experiments of pendulums, as well
as by the experiments of falling bodies. And from all this it
may be concluded, that the refiftanees of bodies, moving in
any fluids whatfoever, though of the moft extreme fluidity^
are, aeteris paribus, as the denfities of the fluids.
Thefe things being thus efiabliflied, we may now determine
what part of its motion any globe projected in any fluid what-
foever would nearly lofe in a given time. Let D be the dia-
meter of the globe, and V its velocity at the beginning of its
motion, and T the time in which a globe with the velocity V
can defcribc in vacuo a fpace that is to the fpace J.D as the
denfity of the globe to the denfity of the fluid; and the globe
projefted in that fluid will, in any other time t, lofe the part
tV TV . .
;rj -9 the part ^p — r~z remaining ; and will deferibe a fpace,
l+t*l+t
which will be to that defcribed in the fame time in vacuo with
the uniform velocity V, as the logarithm of the number
T + t
— == — multiplied by the number 2,30^583093 is to the number
t
^j by cor. 7, prop. 35. In flow motions the refiftance may
be a little lels, becaufe the figure of a globe is more adapted
to motion than the figure of a cylinder defcribed with the
fame diameter. In fwift motions the refifl;ance may be a little
greater, becaufe the elafticity and comprefliion of the fluid
not increafe in the duplicate ratio of the velocity. But
little niceties I take no notice of.
Hex.
SeS.\'tl, OV NATUSAL PHILOSOPHV. 129
And though air. Water, quickfilverj and the like fluids, by
the divifioD of tlieir parts in injinilum, fhould be fubtiiized,
and become mediums iafiDitely fluid, neverlhelefs, the reiifl-
ance they would make to projeded globes would be the fame.
For the refinance coniidered io the preceding propofiliona
arjfes from the inaflivity of the matter ; and the iiiaiftivity of
matter is efleotial to bodies, and always proporlional tu the
quantity of matter. By the divifiou of the parts of the fluid
the refiflauce ariling from the tenacity and fridion of tJie
parts may be indeed diminiihed ; but the quantity of matter
will not be at all diminillied by this diviiion ; and if the quan-
tity of matter be the fame, its force of inaiitivily will be the
fame ; and therefore the refiftance liere fpoken of will be the
fame, as being alwaj-s proportional to that force. To diminilh
this refifl.aoce, the quantity of matter in the fpaces Lhrough
which the bodies move mull be diminiflied ; and therefore;
the celefliat fjiaces, througli which the globes of the planets
and comets are perpetually palling towards all parts, with
the utmoU freedom, and without the leaH fenfible diminutioa
ftbcii- motion, muft be utterly void of any con'oreal Huid>^;
:cepting, perhaps, fome extremely rare vapours and the rays'. I
of light.
Projectiles excite a motion in fluids as they pafa I
them ; and this motion arifes from the excefa of the preflure-'
of the fluid at the fore-parts of the proje^e above the pref-
fure of the fame at the hinder parts ; and caimot be lc& in
ntedxuma infinitely fluid than it is in air, water, and quick-
filver, in proportion to the denfity of matter in each. Now
this excels of prefiure does, in proportion to its quantity, not
only excite a motion in the fluid, but alfo atSls upon the pro-
'"''■''" ^-^ as to retard its motion ; and tliereforc tJie refillaiice
"uid is as the motion excited by the proje(-He io the
'^ 'Cannot be lefs in the moft fublile iether in propor-
'! ,1 ": . ' iliat tether, than it is in air, water,
itMBion to the denQties of Uiofa
real nuiu>.,; i
I the rays'. I
■3 ill rough >\
1G8 MATHEMATICAL PRINCIPLES Book IL
fevcnth foAion^ and confirmed by experiments of falling bo-
dies^ the fame globe in defcribing the fame length would lofe
only apart of its motion equal to , fuppofing the denfity
of water to be to the denfily of air as 8()() to 1. Therefore
the refifianei's were found greater by the* exjKTiments of pen-
dulums (t()r the reafons jull mentioned) tlian by the experi-
ments of falling globes ; and that in the ratio of about 4 to 3.
But yet finee the refiftanees of pendulums ofeillating in air»
water, and c|uieklilver, are alike increafed by like caufes^ the
proportion of the refiftanees in thefe mediums will be rightly
enough exhibited by the experiments of pendulums^ as well
as by the experiments of falling bodies. And from all this it
may be concluded, that the reliftances of bodies, moving in
any fluids whatfoever, though of the moil extreme fluidity^
arc, cateris paribus, as the denfitics of the fluids.
Thefe things being thus eilabliflicd, we may now determine
what part of its motion any globe projected in any fluid what*
foever would nearly lofe in a given time. I^'t D be the dia-
meter of the globe, and V its velocity at the beginning of its
motion, and T the time in which a globe with the velocity V
can deferibe in vacuo a fpace that is to the fpace ^D as the
denfity of the globe to the denfity of the fluid; and the globe
projected in that fluid will, in any other time t, lofe the part
tV TV . .
r|, ? the part , p remaining ; and will deferibe a fpace,
which will be to that deferibed in the fame time in vacuo with
the uniform velocity V, as the logarithm of the number
T + t
— =; — multiplied by the number 2,30238 j093 is to the number
t
^j by cor. 7 9 prop. 35. In flow motions the refiftance may
be a little leis, becaufe the figure of a globe is more adapted
to motion than the figure of a cylinder deferibed with the
fame diameter. In fwift motions tlic refifl:ance may be a little
greater, becaufe the elafticity and compreflTion of the fluid doj
not increafc in the duplicate ratio of the velocity. But th'
little niceties I take no notice of.
128 MATHEMATICAL PRINCIPLES Book IL
feventh fedtion^ and confirmed by experiments of falling bcr-
dies, the fame globe in defcribing the fame length would lofe
only apart of its motion equal to , fuppofing the denfity
of water to be to the denfity of air as 8G0 to 1. Therefore
the rcfifi;anccs were found greater by the experiments of pen-
dulums (for the reafons jufi; mentioned) than by the experi-
ments of falling globes ; <ind that in the ratio of about 4 to 3.
But yet fince the refifi^ances of pendulums ofcillating in air>
water^ and quickfilver^ are alike increafed by like caufes, the
proportion of the refiflances in thefe mediums will be rightly
! enough exhibited by the experiments of pendulums, as well
as by the experiments of falling bodies. And from all thb it
may be conckided, that the refifi;ances of bodies, moving in
any fluids whatfoever, though of the moft extreme fluidity,
are, ceteris paribus, as the denfities of the fluids.
Thefe things being thus eftabliflied, we may now determine
what part of its motion any globe proje<5);ed in any fluid what-
foever would nearly lofe in a given time. Let D be the dia-
meter of the globe, and V its velocity at the beginning of its
motion, and T the time in which a globe with the velocity V
can defcribe in vacuo a fpace that is to the fpace ^-D as the
denfity of the globe to the denfity of the fluid; and the globe
projefted in that fluid will, in any other time t, lofe the part
tV TV . .
rp > the part ^ p , . remaining ; and will defcribe a fpace,
which will be to that defcribed in the fame time in vacuo with
the uniform velocity V, as the logarithm of the number
T + t
— T= — multiplied by the number 2,302585093 is to the number
t
,!kj by cor. 7, prop. SS. In flow motions the refifl:ance may
be a little lefs, becaufe the figure of a globe is more adapted
to motion than the figure of a cylinder defcribed with the
fame diameter. In fwift motions the refiftance may be a little
greater, becaufe the elafi:icity and compreflSion of the fluid do
not incrcafe in the duplicate ratio of the velocity. But thefe
little niceties I take no notice of.
SeS.yiL OF NATURAL PHILOSOPHY. 129
And though air^ WBiet, quickfilver^' and the like fluids, by
the divifion of their parts in infinitum, fhould be fubtilized,
and become mediums infinitely fluids neverthclefs^ the refift-
ance they would make to projedled globes would be the fame.
For the refiftance coniidered in the preceding propofitions
arifes from the inadlivity of the matter ; and the inadivity of
matter is eflential to bodies, and always proportional to the
quantity of matter. By the divifion of the parts of the fluid
the refiftance arifing from the tenacity and fridlion of tlie
parts may be indeed diminifhed ; but the quantity of matter .
will not be at all diminiflied by this divifion ; and if the quan- .
tity of matter be the fame, its force of inadlivity will be the
lame ; and therefore the refiftance here fpoken of will be the
fame^ as being always proportional to that force. To diminifli
this refifl^ance, the quantity of matter in the fpaces through
which the bodies move muft be diminiflied ; and therefore ;
the cdeftial fpaces^ through which the globes of the planets
tod comets are perpetually pafling towards all parts, with
the utmoft freedom, and without the leaft fenfible diminution
of their motion J muft be utterly void of any corporeal fluid,.
excepting, perhaps, fome extremely rare vapours and the rays
of light.
Prqjeftiles excite a motion in fluids as they pafs through
them ; and this motion arifes from the cxcefs of the preflure.
of the fluid at the fore-parts of the projeftile above the pref-
fure of the fame at the hinder parts ; and capnot be lefs in
mediums infinitely fluid than it is in air, water, and quick*,
filver^ in proportion to the denfity of matter in each. Now;
this excefs of preflure does, in proportion to its quantity, not
only excite a motion in the fluid, but alfo a6b upon the pro-!
jedUle fo as to retard its motion ; and therefore the refifl;ance
in every fluid is as the motion excited by the proje<S);ile in the
fluid ; and cannot be lefs in the moft fubtile ssther in propor-
tion to the denfity of that aether, than it is in air, water,
and qnickfilver^ in proportion to the denfities of thofe
fluids.
Vol. ir. K
150 MATHEMATICAL PRINCIPLES Book IL
SECTION VIIL
Of fnotion propagated through Jluid^s..
'• PROPOSITION XLL THEOREM XXXII. ^
Apreffurt is not propagated through a fluid i».rfii£iHinear,di^
reitions, unlejs where the particles of the fluid lie in a right
line. (PI. 8, Fig. 1.)
If 'the pariicies a^ h, Cj d^ e, lie in a right linei tb^ prefl^ie
may be indeed dire<£Uy propagated from a to e; but then the
particle e will urge the obliquely pofiled ^particles f and.jg c^br
liquely) and thofe particles f and g will- not fuftain tbis pf^r
fare^ utiefs they be fupported by the particles h. f^nd k lyiQ^
beyond them ; but Uie particles that fupport ti^m .are f^
preiTed by them ; and thofe particles cannot fuftain tb^t pref-
fure^ Mrithout being fuppcurted by, and preffing upon> thofe
particles that lie ilill farther, as 1 and m, and io on iu ivfi-
fdtum. Therefore the preffure, as fopn as it is propag^ited to
particles that lie out of right lines, begins to defleifl to.wpTd^
one hand and the other, and will be propagated obliquely f^
ifffi/nitum ; and after it has begun to be propagated Qblique)y>
ifi^reachesmore.diflant particles lying out of the right lipc;^
]i will defied); again on each hand ; and this it will do as ofl^a
as it lights on particles that do not lie exadlly in q, right line*
Cor. If any part of a preflure, propagated through a. fluid
from a given point, be intercepted by any obAacle, tti^ re*
raaining part> which is not intercepted, will defied into the
fpaces behind tbe obfts^de. This niay be dcmonfij;at;ed . alfo
after the following manner. Let a prefiure be propagated
from the point A (F]« 8, Fig. £) towards any part, and, if ii
be poffible, in xe<9ilioear .dire&ions ; and the obftacl^ NBCK
being perforated in BC, Jet fill the prefiure be intercept^ but
the coniform part APQ pafling through the circiibr bole BC.
Let the. cone APQ be:divided into fruilifms by the traqiVerfe
planes de, fg, hi. Then while the cone ABC, propagating
the prefiure, urges the conic frufium degf beyond it on the
fuperficics de, and this fruftum urges the next ftufi;um fgihon
the fuperficies fg, and that frufiium urges a third fruftnmi,
^nd fo in infinitum ; it is manifefi; (by the third law) that tbe
Se£t. VIII. OF NATURAL PHILOSOPHY. 131
firft fraftam defg is^ by the re-a6Uon of the fecond fruftum
fghi^ as much urged and preiTed on the fuperficies fg, as it.
uijges and prefTes that fecond fruftum. Therefore the fruftum
degf is comprefTed on both fides^ that is^ between the cone
Ade and the fruftum fliig; and therefore (by cafe 6, prop.
19) cannot preferve its figure^ unlefs it be comprefTed with
the fame force on all fides. Therefore with the fame force
with which it is prefTed on the fuperficies de^ fg^ it will endea-
vour to break forth at the fides df^ eg; and there (being not
in the leaft tenacious or hard^ but perfedlly fluid) it will run
out, expanding itfelf^ unlefs there be an ambient fluid op-
pofing that endeavour. Therefore^ by the effort it makes to
run ont> it will prefs the ambient fluids at its fides df, eg^ with
tbe fame force that it does the fruftum fghi ; and therefore^
the prefTure will be propagated as much from the fides df, eg,
into the fpaces NO^ KL this way and that way^ as it
IS propagated from the fuperficies fg towards PQ. Q.E.D.
PROPOSITION XLII. THEOREM XXXIII.
All motion propagated through a fluid diverges from a reQi^
linear progrefs into the unmoved /paces. (PI. 8, Fig. 3.)
Case l. Let a motion be propagated from the point A
through the hole BC^ and^ if it be poffible^ let it proceed in
the conic fpace BCQP according to right lines diverging
from the point A. And let us firft fuppofe this motion to be
that of waves in the furface of ftanding water; and let de^ fg^
hij kl^ &c. be the tops of the feveral waves^ divided from each
other by as many intermediate valleys or hollows. Then^
becaufe the water in the ridges of the waves is higher than
in the unmoved parts of the fluid KL^ NO^ it will run down
from oflF the tops of thofe ridges e, g, i, ], &c. d^ f^ h^ k^ See.
fhis way and that way towards KL and NO; and becaufe the
water is more deprefTed in the hollows of the waves than in
the i^inmoved parts of the fluid KL, T!iO, it will run down
into thofe hollows out of thofe unmoved parts. By the firft
deflux the ridges of the waves will dilate themfelves this way
and that way, and be propagated towards KL and NO. And
becaufe the motion of the waves from A towards PQ is car-
ried on by a continual deflux from the ridges of the waves into
KS
p
13(2 MATHEMATICAL PRlNCfPLElS Book lU
the hollows next to them^ and therefore cannot be fwiftcr
than iu proportion to the celerity of the defcent; and the
defccnt of the water on each iide towards KL and NO mufl
be performed with tlie fame velocity; it follows, that the dila- '
tation of the waves on each fide towards KL and NO will be
propagated with the fame velocity as the waves themfelvei
go forward with diredly from A to PQ. And therefore the
whole fpace this way and that way towards KL and NO will
be filled by the dilated waves rfgr^ shis^ tklt^ vmnv^^ &c..
Q.E.D. Thatthefe tilings are fo, any one may find by mak-
ing the experiment in ftill water.
Case Q, Let us fuppofe that de^ fg, hi, kl^ nin, rcprefent
pulfes fuccefiively propagated from the point A through an'
elaltic medium. Conceive the pulfes to be propagated by
fucceiTive condenfations and rarefa^lions of the medium, fa
that the denfeft part of every pulfe may occupy a fpherical
fuperficies defcribed about the centre A^ and that equal in-
tervals intervene between the fucccflive pulfes. Let the lines
de, fg, hi, kl, &c. reprefent the denfeft parts of the pulfcis^'
propagated through the hole BC; and becaufe the medium
is denfer there than in the fpaces on either fide towards KL
and NO, it will dilate itfelf as well towards thofe fpaces KL,"
NO, on each hand, as towards the rare intervals between the
pulfes; and thence themedium,becomingalwaysmorerare next
the intervals, and more denfe next the pulfes, will partake of
their motion. And becaufe the progreflSve motion of the
pulfes arifes from the perpetual relaxation of the denfer parts
towards the antecedent rare intervals; and fince the pulfes
will relax themfelves on each hand towards the quiefcent parts
of the medium KL, NO, with very near the fame celerity;
therefore the pulfes will dilate themfelves on all fides into the
unmoved parts KL, NO, with almoft the fame celerity with
which they are propagated diredly from the centre A; and
'therefore will fill up the whole fpace KLON. Q.E.D:
And we find the fame by experience alfo in founds which arc
heard though a mountain interpofe; and, if they come in-
to a chamber through the window, dilate themfelves into all
the parts of the room, and are heard in every comer; £tnd
FlaieyilVSLyoll.
/taff£ 'J3t
yfK*\wmA* ■ *• ATiytWii^h^OMM /*¥W^
. •. \
\ .
-5eS..VnL OF NATURAL PHILOSOPHY. 133
not as refledted from the oppofite walls, but direftly pro^gat-
«d from the window, as far as our fcnfc can judge.
Case 3. Let us fuppofe, laftly, that a motion of any kind is
propagated from A through the hole BC. Then fince the
caufe of this propagation is that the parts of the medium
lliat are near the centre A diflurb and agitate thofe which lie
' farther from it; andfincethe parts which are urged are fluid,
and therefore recede every way towards thofe fpaces where
ihey arc lefs preffed, they will by confequencc recede towards
all the parts of the quicfcent medium; as well to the parts on
■ each hand, as KL and NO, as to thofe right before, as PQ:
and by this means all the motion, as foon as it has pafTed
through the hole BC, will begin to dilate itfelf, and from
thence, as from its principle and centre, will be propagated
diredlly every way. Q.E.D.
PROPOSITION XLIIL THEOREM XXXIV.
Every tremulous body in an elaftic medium propagates the
motion of the pulfes on every fide right forward; but
in a non-elajlic medium excites a circular motion.
Case 1. The parts of the tremulous body alternately
going and returning, do in going urge and drive before them
thofe parts of the medium that lie nearefl, and by that im-
pulfe comprefs and condenfe them; and in returning fuffer
thofe comprefled parts to recede again, and expand them*
felves. Therefore the parts of the medium that lie neareft
to the tremulous body move to and fro by turns, in like man-
ner as the parts of the tremulous body itfelf do; and for the
fame caufe that the parts of this body agitate thefe parts of
the medium, thefe parts, being agitated by like tremors,
will in their turn agitate others next to themfelves; and
thefe others, agitated in like manner, will agitate thofe
that lie beyond them, and fo on in infinitum. And in
the fame manner as the firft parts of the medium were
condenfed in going, and relaxed in returning, fo will
the other parts be condenfed every time they go, and
expand themfelves every time they return. And there-
fore they will not be all going and all returning at the fame
inftant (for in that cafe they would always preferve determined
K 3
136 MATHEMATICAL PRINCIPLES Book II.
celerated or retarded in any place^ as Q^ of a cycloid^ is (by
cor. prop. 51) to its whole weight as its diftance PQ from the
loweft place P to the length PR of the cycloid. Therefore
the motive forces of the water and pendulum^ defcribing the
equal fpaces AE, PQ, are as the weights to be moved ; and
therefore if the water and pendulum are quiefcent at firft,
thofe forces will move them in equal times, and will caufe
them to go and return together with a reciprocal motion.
Q.E.D.
Cor. 1. Therefore the reciprocations of the water in
afcending and defcending are all performed in equal times^
whether the motion be more or lefs intenfe pr remifs.
CoR. 2. If the lengtb of the whole water in the canal be
of &J- feet of French meafure, the water will defcend in one
fecond of time, and will afcend in another fecond, and fo on
by turns in itifinitum ; for a pendulum of 3iV fuch feet in
length will ofcillate in one fecond of time.
Cor. 3. But if the length of the water be increafed or di-
miniflied, the time of the reciprocation will be increafed or
diminiflied in the fubduplicate ratio of the length.
PROPOSITION XLV. THJJOREM XXXVI.
The velocity of z^aves is in the fubduplicate ratio of the
breadth.
This follows from the cpnftruftion of the following pro-
pofition.
PROPOSITION XLVI. PROBLEM X.
Tq find the velocity of waves.
Let a pendulum be conftru6led, whofe length between the
point of fufpenfion and the centre of ofcillatibn is equal to
the breadth of the waves ; and in the time that the pendulum
will perform one fingle qfcillation the waves will advance
forward nearly a fpace equal to their breadth.
That which I call the breadth of the waves is the tranfverfe
meafure lying between tbp deepeft part of the hollows, or
the tops of the ridges. Let ABCDEF (PI. 8, Fig. 5) repre-
fent the furface of ftagnant water afcending and defcending
in fucceffive waves ; and let A, C, E, &c. be the tops of the
waves ; and let B, D, Fj, &c. be the intermediate hollows.
lin/^nim/E. „^,a6
134. MATHEMATICAL PRINCIPLES ' Book IL
diftances from each other^ and there could be no alternate,
condenfation and rarefadlion); but fince^ in the places where
they are condenfed^ they approach to^ and^ in the places
ivhere they are rarefied^ recede from each other^ therefore
fome of them will be going while others are returning; and
, fo on in infinitum. The parts fo goings and in their going
condenfedj are pulfes^ by reafon of the progreiliYe motion
with which they ftrike obftacles in their way ; and therefore
. the fucceiBve pulfes produced by a tremulous body will be
propagated in reAilinear diredlions ; and that at nearly equal
diftances from eacli other^ becaufe of the equal intervals of
time in which the bpdy^ by its feveral tremors^ produces the
feveral pulfes. And though the parts of the tremulous body
go and return in fome certain and determinate dire6lion^ yet
the pulfes propagated from thence through the medium will
dilate themfelves towards the fides^ by the foregoing propo-
fition ; and will be propagated on all lides from tliat . tremu-
lous body, as from a common centre, in fuperficies niearly
. fpherical and conqentrical. An example of this we have in
,^ waves excited by fhaking a finger in water, which proceed
not only forwards and backwards agreeably to the motion of
the finger, but fpread themfelves in the manner of concen-
. tripal circles all round the finger, and are propagated on every
fide. For the gravity of the water fupplies the place of elaflic
force.
Case 2. If the medium be not elaftic, then, becaufe its
parts cannot be condenfed by the prefTure arifing from the vi-
brating parts of the tremulous body, the motion will be pro-
, pagated in an inftant towards the parts where the medium
yields moft eafily, that is, to the parts which the tremulous
body would otherwife leave vacuous behind it. The cafe is
the fame with that of a body projected in any medium what-
ever. A medium yielding to projeAiles does not recede in in^
, finitum, but witli a circular motion comes round to the fpaces
which the body leaves. behind it. Therefore as oflen as a tre-
mulous body tends to any part, the medium yielding to it
comes round in a circle to the parts which the body leaves ;
and as oflen as the body returns to the firft place, tke me-
Flale.ysxyoli. /ua/fiiM
C
Mrytwnum *cM/. WM * fm » f *tri^ Jlyk.y=
Se3. Vin. or natural philosophy* 135
dium will be driven from the place it came round to^ and re-
turti to its original place. And though the tremulous body
*hd not firm and hard^ but every way flexible^ yet if it con-
tinue bf a given magnitude^ fince it cannot impel the medium
by its tremors any where without yielding to it fomewhere
elfe^ the medium receding from the parts of the body where
it is prefled will always come round in a circle to the ports
that yield to it. Q.E.D.
Cor; It is a miftake^ therefore^ to think, as fome have done,
that the agitation of the parts of flame conduces to the propa-
gation of isi prelTare in rectilinear dirediions through an am-
bient mediuni'. A prefTure of that kind muft be derived not
from the agitation only of the parts of flame, but from the
' 'dilatation Of the whole.
PROPOSITION XLIV. THEOREM XXXV.
If water afcend and defccnd alternately in the ereSed legs
l^L, MN, of a canal or pipe ; and a pendulum be conJlruSed
whofe length between the point offufpenfion and the centrt of
ofcillation is equal to half the length of the water in the ca»
nal; I fay, that the water will afcend and defcend im the
fame times in ichich the pendulum ofcillates. (PI. 8, ^. 4.)
I meafure the length of the water along the axes of the
canal and its legs, and make it equal to the fum of thofe axes ;
and take no notice of the refiftance of the water ariiing from
its attrition by the fides of the canal. Let, therefore, AB, CD,
reprefent the mean height of the water in both legs ; and when
the water'in the leg KL afcends to tlie height EF, the water
will defcend in the leg MN to the height GH. Let P be a
pendulous body^ VP the thread, V the point of fufpenfion,
RPQS the cycloid which the pendulum defcribes, P itsloweft
point, PQ an arc equal to the height AE. The force uTith
which t]be motion of the water is accelerated and retarded al-
ternately is the excefs of the weight «f the water in one leg
above the weight in, the other ; and, therefore^ when the water
in the leg KL afcends to EF, and in the other leg defcends
to GH, that force is double the weight of the water EABF,
arid therefore is to the weight of the whole water as AE or PQ
to VP or PR. The force alfo with which the body P is ac-
K 4
r
136 MATHEMATICAL PRINCIPLES Book II.
^ celerated or retarded ia any place^ as Q^ of a cycloid, is (by
cor. prop. 51) to its whole weight as its diftance PQ from the
loweft place P to the length PR of the cycloid. Therefore
' the motive forces of the water and pendulum, defcribing the
equal fpaccs AE, PQ, are as the weights to be moved; and
therefore if the water and pendulum are quiefcent at firft,
thofe forces will move them in equal times, and will caufe
them to go and return together with a reciprocal motion.
Q.E.D.
Cor. 1. Therefore the reciprocations of the water in
. afcendiug and dcfcending are all performed in equal times^
whether the motion be more or Icfs intenfe or remifs.
CoR. 2. If the length of the whole water in the canal be
of G\ feet of French meafure, the water will dcfcend in one
fecond of time, and will afcend in another fecond, and fo on
by turns in infijiiium ; for a pendulum of 3iV fuch feet in
length will ofcillate in one fecond of time.
CoR. 3. But if the lengtb of the water be increafed or di-
• minifhed, the time of the reciprocation will be increafed or
diminiflied in tlie fubduplicate ratio of the length.
PROPOSITION XLV. THJiOREM XXXVI.
The velocity of waves is in the fubduplicate ratio of the
breadths.
This follows from the cpnftruftion of the following pro-
portion.
PROPOSITION XLVI. PROBLEM X.
Tq find the velocity of waves.
Let a pendulum be conftru6led, whofe length between the
point of fufpenfion and the centre of ofcillatibn is equal to
the breadth of the waves ; and in the time that the pendulum
will perform one fingle qfcillation the waves will advance.
forward qearly a fpace equal to their breadth.
That which I call the breadth of the waves is the tranfverfe
meafure lying between thp deepeft part of the hollows, or
the tops of the ridges. Let APCDEF (PI. 8, Fig. 5) repre-
fent the furface of flagnant water afcending and defcending
in fucceffive waves ; and let A, C, E, &c. be the tops of the
waves ; and let B, D, Fj^ &c. be the intermediate hollows.
I'la/^nim/II. n^„,M
«^
-°.>
"^
--^-
-^
5 ^
A B r
D
11
v,_
_y
ne- \
I.
K
Se3. VIII. OF KATUKAL PHILOSOPHY. 137
Becaufe the motion of the waves is carried on by the fuccef-
five'afcentand defcentof the water^ fo that the parts thereofj
as A^ C^ E^ &c. which are higheft at one time become lowed
immediately after ; and becaufe the motive force^ by which
the higheft parts defcend and the loweft afcend^ is the weight
of the elevated water/ that alternate afcent and defcent will
be analogous to the reciprocal motion of the water in the
canalj and obferve the fame laws as to the times of its afcent
and defcent ; and therefore (by prop. 44) if the diftances be-
tween the higbeft places of the waves A, C, E, and the loweft
B, D, F, be equal to twice the length of any pendulum^ the
higheft parts A^ C^ E^ will become the loweft in the time of
one ofcillation^ and in the time of another ofcillation will
afcend again. Therefore between the paftage of each wave
the time of two ofcillations will intervene ; that is^ tlie wave
wUl defcribe its breadth in the time that pendulum will ofcil-
late twice ; but a pendulum of four times that lengthy and
which therefore is equal to the breadth of the waves, will juft
ofcillate once in that time. Q.E.I.
Cob. 1. Therefore waves, whofe breadth is equal to 3-^
French feet, will advance through a fpace equal to their
breadth in one fecond of time ; and therefore in one minute
will go over a fpace of 183^ feet ; and in an hour a fpace of
1 1000 feet, nearly.
Cob. 2. And the velocity of greater or lefs waves will be
augmented or diminiftied in the fubduplicate ratio of their
bread til.
Thefe things are true upon the fuppofition that the parts
of water afcend or defcend in a right line ; but, in truth, that
afcent and defcent is rather performed in a circle ; and there-
fore I propofe the time defined by this propofition as only
near the truth.
PROPOSITION XLVII. THEOREM XXXVII.
Tfpulfes are propagated through a fluid, thefeveral particles
of the fluid, going and returning with thefliorteft reciprocal
motion, are always accelerated or retarded according to thp
law of the ofcillating pendulum. (PL 9> Fig« !•)
>
i38 MATHEMATICAL rRINClP£.ES jBcN>Jk IL
Let AB^ BC, CD, See. reprefent equal diftances of fuccef-
. five pulies ; ABC the line of direclion of the motion of the
. facceffive pulfes propagated from A to B; E, F, 6 three
. pbyfical points of the quiefcent medium lituate in the right
line AC at equal diftances from each other; Ee^ Ff, Gg,
equal fpaces of extreme ihortnefs^ through which thofe points
go and return with a reciprocal motion in each vibration ; i,
09 y* Ai^y intermediate places of the fame points ; EFj FG
phyfical lineols^ or linear parts of the medium lying between
tbofc points^ and fucccflively transfcred into the places i^,
(Py, and ef, ^ fg. I^t there be drawn the right line
equal to the right line Ee. BifeA the fame in O, and from
the centre O, with the interval OP. dcfcribe the circle SlPi
Let the whole time of one vibration, with its proportionally
parts, be expounded by the whole circumference of this cir-^
cle and its parts, in fuch fort, that, when any time PH of
PHSh is completed, if there be let fall to PS tlie perpendicu-'
lar HL or hi, and there be taken E; equal to PL or PIj the
phyiical point E may be found in & A point, as E, moving ac-'
cording to this law with a reciprocal motion, in its goings
from E through £ to e, and returning again through £to E,
will perform its fevcral vibrations with the fame degrees of aC"
celeration and retardation with thofe of an ofcillating pendu-
lum. We are now to prove that the feveral phyfical points
of the medium will be agitated with fuch a kind of motion^
Let us fuppofe, then, that a medium hath fuch a motion ex —
cited in it from any caufe whatfoever, and coniider whaSs
will follow from thence.
In the circumference PHSh let there be taken the equ&T
arcs HIjIK, or hi, ik, having the fame ratio to the whole cir
cumference as the equal right lines EF, FG have to BCj th
whole interval of the pulfes. Let fall the perpendiculsurs
KN, or im, kn ; then becaufe the points E, F, 6 are fuqcefliv<
ly agitated with like motions, and perform their entire vib;
tions compofed of their going and return, while the pulfe
transfered from B to C ; if PH or PHSh be the time elapfe
iincc the beginning of the motion of the point £, then wi
PI or PHSi be the time elapfed fincc the beginning of t
y
S^S.VIII. OP KATURAL PHTLOSOPHY. 139
siottoti of the point F^ and PK or PHSk the time elapfed
£nce the beginning of the motion of the point 6 ; and there-
fore Eft Ffl), Gy, will be refpeaively equal to PL, PM, PN,
*while the points are going, and to PI, Pm, Pn, when the
^ints are returning. Therefore £y or EG + Gy — E« will,
*when the points are going, be equal to EG — LN, and
in their ristum equal to EG + In.. But ey is the breadth or
expatifion of the part EG of the medium in the place ty ;
ond therefore the expanfion of that part in its going is to its
- * JDleaa expanfion as EG — LN to EG ; and in its re-
•tuni, as EG + In or EG + LN to EG. Therefore
fince LN is to KH as IM to the radius OP, and KII to
'EG as the circumference PHShP to BC ; that is, if we put
V' foir the radius of a circle whofe circumference is equal to
BC the interval of the pulfes, as OP to V ; and, tx taquo, LN
to' EG as IM to V ; the expanfion of the part EG, or of the
' phyfical point F in the place ey, to the mean expanfion of
'' -the&me part in its firft place EG, will be as V — IM to V
* ii&^ing, and as V + im to V in its return. Hence the elaf-«
tic force of the point F in the place sy to its mean elaftic
force in the place EG is as ^ ,^ to ?? in its going, and as
«— T— tOT7 in its return. And by the fame reafoning the
* efafiic forces of the phyfical points E and G in going
we as Tf — TTT and t? j^r— to ^7 ; and the difference of
V — HL V — KN V '
the forces to the mean elaftic force of the Inedium as
HIr-KN ■ 1 !_ . HL— KN
VV^VkHL-VxKN+HLxKn'^v;*^^*"'^ VV
to T?9 or as HL — KN to V ; if we fuppofe (by reafon of the
very Ihort extent of the vibrations) HL and KN to be indefi-
nitely lefs than the quantity V. Therefore fince the quan«
tity V is given, the difference of the forces is as HL — KN ;
' that is (becaufe HL — KN is proportional to HK, and OM
to 01 or OP; and becaufe HK and OP are given) as OM ;
thai is, if Ff be bifedted in Q, as Q0. And for the fame
leafon the difference of the elaftic forces of the phyfical
points £ and y, in the return of the phyfical lineol® e^^ is as
140 MATHEMATICAL PRINCIPLES jBooik lE^ II
Q^. Rut that difference (that is, the excefs of the rliill "MHIl
force of the point e above the elaftic force of the point y) is
the very force by which the intervening phyiical lineolae
of the medium is accelerated in going, and retarded in
turning ; and therefore the accelerative force of the phyfi<
lineolae ey is us its diilancc from Q, the middle place of tl
vibration. Therefore (by prop. 38, book 1) the time is right-
expounded by the arc PI*; and the linear part of the mediozi^sn
iy is moved according to the law above-mentioned, that SSSf
according to the law of a pendulum ofcillating; and the ca ^^e
is the fame of all the linear parts of which the whole medh
is compounded. Q.E.D.
Cob. Hence it appears that the number of the pulfes
pagated is the fame with the number of the vibrations of
tremulous body, and is not multiplied in their progrefs.
the phj'fical lineolaB ey as foon as it returns to its firft
is at reft ; neither will it move again, unlefs it receives a nfc."^^
motion either from the impulfe of the tremulous body, or ^cdI
the pulfes propagated from that body. As foon, therefore, ^Stf
the pulfes ceafe to be propagated from the tremulous body, it
will return to a ftate of reft, and move no more.
PROPOSITION XLVIII. THEOREM XXXVIH.
T/ie velocities of pulfes propagated in an elajtic fluid arc in *
ratio compounded oftkefubduplicate ratio of the elafi' *^
force direiilrfy and the fubduplicate ratio oftlic dtnfity ti
verfelif; fuppofing the elaftic force of the fluid to he
portional to its condenfation.
Case 1. If the mediums be homogeneous, and the di
tances of the pulfes in thofe mediums be equal amongft thei
fclves, but the motion in one medium is more intenfe
in the other, the contradlions and dilatations of the corre-^***
pondent parts will be as thofe motions : not that this pi^' "^
portion is perfedlly accurate. However, if the contra&ior — ==^*
and dilatations are not exceedingly intenfe, the error wi ;
not be fenfible ; and therefore this proportion may be pool
dered as phyfically exa6l. Now the motive elaftic forces
as the contradlions and dilatations ; and the velocities gent
rated in the fame time in equal parts are as the force-
eS. Vlil. OF NATURAL PHILOSOPHY. US
if the liiieblae ; and therefore, ex tequo, the foi^e with whicli
.he lineolss EG is urged in the places P and S is to the
g^eight of that lineolse as HK x A to V x EG; or as PO x A
to VV; becaufe HK was to EG as PO to V. Therefore fmce
Ihe times in which equal bodies are impelled through equal
fpaoes are reciprocally in the fubduplicate ratio of the forces^
loe time of one vibration^ produced by the adlion of that elafiic
fbroej win be to the time of a vibration^ piroduced by the im-
pnlfe of the weight in a fubduplicate ratio of VV to PO x A\
and therefore to the time of the ofcillation of a pendulum
w)idfe length is A in the fubduplicate ratio of VV to PO x
Ik, and the fubduplicate ratio of PO to A conjundlly; that
isjiii the entire ratio of V to A. But in the time of one vibration
coinpbfed of the going and returning of the penduluni^ the
pulle will be propagated right onwards through a fpace e(][ual
to its breadth BC. Therefore the time in which a pulfe run.<
over the fpace BC is to the time of one ofcillation compofed
of fSie going and returning of the pendulum as V to A, thai
I9« as BC to the circumference of a circle whofe radius is A.
But the time in which the pulfe will run over the fpace BC
is to the time in which it will run over a length equal to that
circumference in the fame ratio; and therefore in the time
of fuch an ofcillation the pulfe will run over u length equal
to that circumference. Q.E.D.
Cob. 1. The velocity of the pulfes is equal to that which
heavy hodies acquire by falling with an equally accelerated
liiotion^ and in their fall defcribing half the altitude A. For
the pulfe will^ in the time of this fall^ fuppoiing it to move
with the velocity acquired by that fall^ run over a fpace that
will be equal to the whole altitude A ; and therefore in the
time of one ofcillation compofed of one going and return^ will
so over a fpace equal to the circumference of a circle defcrib-
ed with the radius A ; for the time of the fall is to the time
of ofcillation as the radius of a circle to its circumference.
Cob. 2. Therefore (ince that altitude A is as the elaftic
jforce of the fluid diredly, and the denfity of the fame ihverfe-
Ij, the velocity of the pulfes will be in a ratio compounded
144 MATHEMATICAL PRINCIPLES £ook IC
of the fubduplicatc nitio of the denfity inverfely^ and th^
fubduphcate ratio of the claftic force diredll}'.
PROPOSITION L. PROBLEM XIL
To find the dijiances ofthepulfes.
Let the number of the vibrations of the body, by whof^CTfci
tremor the pulfes are produced, be found to any given time. ^^^-^
By that number divide the fpace ^vhich a pulfe can go oveiK
in tlic fame time, and the part found will be the breadth ofc
one pulfe. Q.E.L
SCHOLIUM.
Tlie lait propofitions refpe6l the motions of Ught and founds ;£
for fince hght is propagated in right lines, it is certain that
it cannot coufifl in adion alone (by prop. 41 and 4d). As to
founds, fince they aiife from tremulous bodies, they can be
nothing elfe but pulfes of the air propagated through it (by
prop. 43); and this is confirmed by the tremors which
founds, if they be loud and deep, excite in the bodies near
them, as we experience in the found of drums; for quick
and fhort tremors are lefs eafily excited. But it is well
known that any founds, falling upon firings in unifon with
the fonorous bodies, excite tremors in thofe fi;rings. This
is alfo confirmed from tlie velocity of founds ; for fince the
fpccific gravities of rain-water and quickfilvcr are to one
another as about 1 to I3y, and when the mercury in the
barometer is at the height of SO inches of our meafure, the
fpecific gravities of the air and of rain-water are to one ano-
ther as about 1 to 870, therefore the fpecific gravity of ait
and quickfilver are to each other as 1 to 1 1 890. Therefore
when the height of tlie quickfilver is at SO inches, a height
of unifonn air, whofe weight would be fufficient to compreli
our air to the denfity we find it to be of, mufl; be equal t6
356700 inches, or £9725 feet of our meafure : and this is
that very height of the medium, which I have called A in
the conftru6ljon of the foregoing propofition. A circle whofe
radius is 29725 feet is I86768 feet in circumference. And
fince a pendulum S9f inches in length completes one ofcilla-
tion, compofed of its going and return, in two feconds of
time, as is commonly k^own, it follows that a pendulum
SeS.WlU. OF NATVBAL PHILOSOPHY. 145
29785 feet^ or 356700 inches in length ^ill perform a like
ofcillation in 190| feconds. Therefore in that time a found
will go right onwards 1 66768 feet^ and therefore in one fe-
<x>nd 979 feet.
But in this computation we have made no allowance for
the craflitude of the folid particles of the air^ by which the
found is propagated inilantaneoufly. Becaufe the weight
of air is to the weight of water as 1 to 870^ and becaufe falts
are almoft twice as denfe as water; if the particles of air are
foppofed to be ofneaip.the fame denfity as thofe of water
or {elty and the rarity of tfie air arifes from the intervals of
the particles; the diameter of one particle of air will be to the
interval between the centres of the particles as 1 to about
9 or 10^ and to the interval between the particles themfelves
as 1 to 8 or 9. Therefore to 979 feet^ which^ according to
the above calculation^ a found will advance forward in one
fecond of time^ we may add V^^, or about 109 feet^ to com-
penfate for the craffitude of the particles of the air: and then
a found will go forward about 1088 feet in one fecond of time.
Moreover^ the vapours floating in the air being of another
fpring, and a different tone^ will hardly^ if at all^ partake of
the motion of the true air in which the founds are propagated.
Now if thefe vapours remain unmoved^ that motion will be
propagated the fwifter through the true air alone^ and that
in tlie fubduplicate ratio of the defe6i of the matter. So if
the atmofphere confift of ten parts of true air and one part
of vapours^ the motion of founds will be iWifter in the fbb-
duplicate ratio of 11 to 10^ or very nearly in the entire ratiD
of 21 to £0; than if it were propagated through eleven parts
of true air: and therefore the motion of founds above dif-
covered muft be increafed in that ratio. By this means the
found will pafs through 1142 feet in one fecond of time.
Thefe things will be found true in fpring and autumn^
when the air is rarefied by the gentle warmth of thofe feafons^
and by that mean^ its elaftic force becomes fomewhat more
intenfe. But in winter^ when the air is condenfed by the
cold^ and its elafiic force is fomewhat remitted^ the motion
Vol. IL L
I
146 MATHEMATICAL PSiKCirLM Book IL
of founds will be flower in a fubduplicate ratio of the denfi^ ; <--
and^ on the other hand, fu^fter in the fummer.
Now by experiments it a&ually appears that fimads do ^^. j
really advance in one fecond of time about 1 142 feet .of J&l^«--^s^^
glf/h meafure, or 1070 feet of French meafure.
The velocity of founds being known, the intervals of thi
pnlfes are known alfo. For M. Saieveur, by fome expeii-^
ments that he made, found that an open pipe about five
feet in length gives a found of the fame tone with a viQI-ftrin^
that vibrates a hundred times in one fecond. Therefore th
are near 100 pulfes in a fpace of 1070 Paris feet, which
found runs over in a fecond of time ; and therefore one pol
fills up a fpace of about lOi^ Paris feet, that is, about twii
the length of the pipe. From whence it is probable that
breadths of the pulfes, in all founds made in open pipes,
equal to twice the length of the pipes.
Moreover, from the corollary of prop. 47 appears the
fon why the founds immediately ceafe with the motion of
the fonorous body, and why they are heard no longer wfaeo
we are at a great diftance from the fonorous bodies than when
we are very near them. And befides, from the £oregoim
prmciples, it plainly appears how it comes to pafs that foundi
are fo mightily increafed in fpeaking-trumpets ; for all re-
' ciprocal motion ufes to be increafed by the generating caufe
at each return. ' And in tubes hindering the dilatation of tbe
founds, the motion decays more flowly, and recurs more
forcibly ; and therefore is the more increafed by the new mo-
tion impreffed at each return. And thefe are the principal
phsenomena of founds.
SECTION IX.
Of the circular motion of fluids.
HYPOTHESIS.
The rejijlance arijingfrom the want of lubricity in the parts
of a fluid, is, ca^teris paribus, proportional to the velocitff
with which the parts of the fluid are fepardted from eac&
other.
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SfS. tX: OP NATT7RAL PHILOSO?HY. 14?
PROPOSITION LI. THEOREM XXXVIIL
£^ M JbKd cylinder ir^nitefy long, in an uniform and ir^nitt
Jbdd, revolve with an uniform motion about amtxii given in
pofiiion, eind the fluid be forced round by only this impulft
^thc eyiinder, and every part of the fluid perfevere uni"
fptndy in its motion ; I fay 9 thai the periodic times of the
parts of the fluid are >as their diftances from the ax^is of the
yUnder.
Let AFL (PI. g^ Fig. 2) be a cylinder turning uniformly
aboBt llie axis S> and let the concentric circles BGM^ CHN^
WOy JSXPf &c.. divide the fluid into innnmerable concen-
tric cylindric folk! orbs of the fame thicknefs. 3%ten^ becanfe
the fluid is homogeneous^ the impreflions which the conti-
gVOUB orbs make upon each other mutually will be {by the
hy)>othefis) as their traxiflations from each otber^ and as the
oontigaous fuperficies upon which the impreffions are made*
If the impreflion made upon any orb be greater or lefs on its
ooncUve than on its convex fide, the ftronger impreffion wlH
prevail^ and will either accelerate or retard the motion of the
oAj according as it agrees with^ or is contrary to^ the motion,
oflbefame. Therefore, that every orb may perfevere uni-
fennly in its motion, the impreffions made on both' fides
■mil 1)e ^equal, and their dirediions contrary. Tlierefore fince
the impreffions are as the contiguous fuperficies^ and as that
tranflations from one another, the tranflations will be in-
verfely as the fuperficies, that is, inverfely as the diftances of
tbe fuperflcies from the axis. But the differences of the an-
gular motions about the axis are as thofe tranflations applied
to the diftances, or as the tranflations diredlly aadthedift^
aaces inverfely ; that is, joining thefe ratios together, as the
fqaares of the difl^nces inverfely. Therefore if there be
ereAed the lines Aa, Bb, Cc, Dd, Ee, 8cc. perpendicular to
the feveral parts of the infinite right line SABCDEQ, aiid re-
ciprocally proportional to the fquai^es of SA, SB, SC, SD, S£^
8cc. and through the extremities of thofe perpendiculars there
be fappofed to pafs an hyperbolic curve, the fums of thedif-
feveoees, that is, the whole angular motions, will be as the
coixefpoadent fums of the Jines Aa,^ Bb^ Cc^ Di, £e, that
L 2
-f »r
ir.t«T
i* ' ::' tt rmiriiui- l ni?diiiiL uiiii:nz:"i &zid the smiber of'
f:.:tfi/. at'Uit vyitiryAn arear .-^i*^.. BdQ. CrQ, DdQ^EK},
tLi. s^Uiiinr'jiir i: UK iiim? saiL ztif Tiinf^. reciprocalT pn>-
yjTUiOitL i: i!K ucrum: niriiinii?^ vM tit iilfc redpnxsfljfnH
ffunrjuiL. i: trmk ti7«:&<^ T:A-::Tt:33*» utt jiecicidic tmeof any
riii-H... br I* ir r£:rniraaa-T a* iiit u7E:ft.IldQ, iliai is (asqi-
rt zr-jiL "lit* i:ii:*vT, nH-Uiniir o: nniiarBlnr** of coz^vci^di-
rK;" r 21- "Xj* arfciiTi'** ^Ti C* Z. I*
Cm. _ . H«:ii:« lut uifiiubr juouoxtf uf ir«e porticla of die
£l^i bj-i r&::jp:".traJ:j( ut Uieir nTraTiPe* iram tbe axk of tte
CrlinOil-. SDC lilt LliiaiUIf 1=£jDCJII£:» LreeoQ&L
Cc £. £. If L £lic lit f »TiTiiiin*f ii: £ cv" JTicric veflU of id
irriLi^ jenrii.. lii£ cxxruor Likzr.lkis crtaaei vithiDy and
bo^ Uie c-v.^de:r? rrrccrt s-iioci os^ cammcm axs. and the
pan cc il>r rcjd perferere^ in .i^ id9d:>zi. ibe periodic tinei
of lie frreriL pic-i* wl^ ie 8a iiit ci^&dok from the axis of
like cxliDdens^
Cojc, 5. If ibere be added c«r TLsifE a-^rav anv commaB
cuac'JiY of arriJir nioDr fr:a :"re crliDdfr arid flaid mov-
ins in ihn miLZZj^T: ve- r»tcii::t ir.iii sfw moiion will not
alicT the mL:;:sI aiirliioD c: iLe r^ferij v'f ibe ndd. the motioa
of tbe pan3 azson^r iheaafc'ivcs v^ 2M bf changed ; for tlie
traDilatioDs of :hc pans irr'zi one 2j>oiber depend upon the
attrilion. Anr pan will p^nerere in that isotion^ wbicbj
by tbe^tirition made on both fcdes with contrary diie^onSi
is no more accekr&ted than h i» reianded.
Cob. 4. Therefore if there be t^ken awav from this whok
fyfiem of the cylinders and the £uid all the angolar motion
of the outward cylinder, we (hall have the motion of the fluid
in a quiefcent cylinder.
Cor. 5. Therefore if the fluid and outward cyhnder are at
reft, and the inward cylinder revolve uniformly, there will be
communicated a circular motion to the fluid, which will be
propagated by degrees throng the whole fluid ; and will go
on continually increaCng, till foch time as the fevenil parts
of the fluid acquire the motion determined in cor. 4*
ScB. IX. 07 NATURAL PHILOSOPHY. 149
Cor. 6. And becaufe the fluid endeavours to propagate its
motion ftill farther^ its impulfe will carry the outmoft cylinder
alio about with it^ unlefs the cylinder be violently detained ;
and accelerate its motion till the periodic times of both cy-
linders become equal among themfelves. But if the outward
cylinder be violently detained^ it will make an effort to retard
the motion of the fluid ; and unlefs the inward cylinder pre-
ferve that motion by means of fome external force impreffed
thereon^ it will make it ceafe by degrees.
An theie things will be found true by making, the expe-
riment in deep ftanding water.
PROPOSITION LIL THEOREM XL.
If a fotid/phercy in an uniform and infinite fluid, revolves
about an axis given in pofition with an uniform motion, and
the fluid be forced round by only this impulfe of thefphere ;
and every part of the fluid perfeveres uniformly in its motion:
Jfoy, that the periodic times of the parts of the fluid are as
thefquares of their diftancesfrom the centre of thefphere.
Cabb 1. Let AFL be a fphere turning uniformly about the
axis Sj and let the concentric circles BGM^ CHN^ DIO^
EKPj &c. divide the fluid into innumerable concentric orbs
of the (ame thicknefs. Suppofe thofe orbs to be folid; and^
liecanfe the flaid is homogeneous^ the impreffions which the
oontignous orbs make one upon another will be (by the fup«
pofition) as their tranflations from one another^ and the con-
tigBOUS fuperficies upon which the impreffions are. made. If
the impreffion upon any orb be greater or lefs upon i(s con-
cave than upon its convex flde^ the more forcible impreflion
will prevail^ -and will either accelerate or retard the velocity
of the orbj according as it is dire6led with a confpiring or
contrary motion to that of the orb. Therefore that every orb
may perfevere uniformly in its motion^ it is neceffary that the
impreffions made upon both fides of the od) fliould be equals
and have contrary diredlions. Therefore fince the impreflions
are as the contiguous fuperficies^ and as their tranflations
from one another^ the tranflations will be inverfely is the fu-
perficies^ that is^ inverfely as the fquares of the diflances of
the fuperficies from the centre. But the diflerences of the
angular motions about the axis are as thofe tranflations ap-
L 3 V
150 MATHEMATICAL PBINCIPLfiS BooklL
plied to the didances^ or as the banflations dire&ly and the
diftances inverfely ; that is, by compoundiDg thofe ratios, as
the cubes of the diftances inverfely. Tberetbre if upon the
fevcral parts of the infinite right line SABCDEQ there be
ere&ed the |>erpendiculars Aa, ]ih, Co, Dd, Ee^ &c. recipro*
cally proportional to the cubes of SAj SB^ SCj SD, SE, 8lc»
tlie fuisis of the differences^ that is, the whole angular mo-
tions, will be as the correfponding fuins of the lines Aa^ Bb,
Cc, Dd, £c, &c. that is (if to confiitute an uniformly fluid
medium tlie number of the orbs be increafed and their thick-
nefs dimini(hed in injimtum)^ as the hyperbolic areas AaQ^
BbQ, CcQj DdQ, EeQ, 8cc. analogous to the fums ; and the
I)eriodic times being reciprocally proportional to the angular
motiooB, will be alio reciprocally proportional to thofe aceas.
Therefoixi die periodic time of any orb DIO is reciprocally
as the area DdQ, that is (by the known methods of quadra-
tures), direAly as the fquare of the diftance SD. Which was
£rfl to be demonftrated.
Case (2. From the centre of the fphere let there be drawn
a great number of indefinite right lines, making given angles
with the axis, exceeding one another by equal diifereoces;
and, by thefe lines revolving about the axis, conceive the orbs
to be cut into innumerable annuli ;. then will every annulus
have four annuli contiguous to it, that is, one on its infideL
one on its outfide, and two on each band. Now each; of
thefe annuU cannot be impelled equally and with ecwtrary di-
reAions by the attrition of the interior and exterior annuli,
unlefs the motion be communicated according to the law
which we demonftrated in cafe 1. This appears from that,
demonftradon. And therefore any feries of annuli, taken in
any right line extending itfelf in ir^mtum from the globe^
will move according to the law of cafe 1, except We ihould
imagine it hindered^by the attrition of the annuli on each fide
of it. But now in a motion, according to this law, no fuch
attrition is, and therefore cannot be, any obftacle to the mo-
tions perfevering according to that law. If annuli at equal
diflances from the centre revolve either more fwiftly or more
flowly near the poles than near the ecliptic, they will be acr
ceierated if flow, and retarded if fwift, by dieir mutual attri*
&A« IX. OP NATURAL PHILOSOPHY. 151
tfion ; mod lb the periodic times will continually approach to
eqoalitjj according to the law of cafe 1. Therefore this at-
tritioQ will not at all hinder the motion from going on ac-
Qordii^ to the law of cafe 1^ and therefore that law will take
]dace ; . that is, the periodic times of the feveral annuli will be
at the fquares of their diftances from the centre of the globe.
Whicii was to be demonftrated in the fecond place.
Cas A 3. Let now every annulus be divided by tranfverfe
fedions into innumerable particles conftituting a fubftance ab-
fidotely and uniformly fluid ; aiid becaufe thefe fe&ions do
■ot at all refpedl the law of circular motion^ but only ferve
to piodaCe a fluid fubfl^nce^ the law of circular motion will
eoBtinue the fame as before. AH the very fmall annuli will
eidwr not at all change their afperity and force of mutual at-
trition upon account of thefe fe6):ions^ or elfe they will change
die fame equally. Therefore the proportion of the caufes re-
maining the fame^ the proportion of the efle^ will remain
ite lame alfo ; that is^ the proportion of the motions and the
•periodic times. Q.E.D. But now as the circular motion,
and the centrifugal force thence ariflng, is greater at the
idiptic than at the poles, there mufl: be fome caufe operating
to Ktein the feveral particles in their circles ; otherwife the
aiatttt that is at the ecliptic will always recede from the cen-
tli^, and come round about to the poles by the outfide of the
tortex, and from thence return by the axis to the ecliptic
with a perpetual, circulation.
Cos. 1. Hence the angular motions of the parts of the
imd about the axis of the globe are reciprocally as the
£|nareB of the difliances from the centre of the globe, and the
^ib£blute velocities are reciprocally as the fame fquares applied
to th^ diflances from the axis.
CSos. £. If a globe revolve with a uniform motion about an
axis of m given pofition in a fimilar and infinite quiefcent fluid
with an uniform motion, it will communicate a whirling mo*
tion to the fluid like that of a vortex, and that motion will
hy diegrees be propagated onwards in infinitum ; and this mo-
tioD will be increafed continually in every part of the fluid,
L4
132 MATHEMATICAL FRINCIFXIS BooklL
uH the periodical times of the feveral parts become as the
fquares of the diftances from the centre of the globe.
Cor. 3. Becaafe the inward parts of the vortex are by rea^
fon^of their greater velocity continually preffing upon and
dri\'ing forwards the external parts, and by that adion-are
perpetually communicating motion to them, and at the fame
time thofe exterior parts communicate the fame qaantity of
motion to thofe that lie ftill beyond them, and by this
action prefer\'e the quantity of their motion continnally on-
changed, it is plain that the motion is perpetnally transiiered
from the centre to the circumference of the vortex, ^1 it b
quite fwallowcd up und loft in the boundkis extent of that
circumference. The matter between any two fpherical fuper-
ficies conceiitrical to the vortex will never be acc^elerated ;
becaufe that matter will be always transfering the motion it
receives from the matter nearer the centre to that matter
which lies nearer the circumference.
Cob. 4. Therefore, in order to continue a vortex in thefame
(late of motion, fome adiive principle is required from which
the globe may receive continually the fame quuitity of mo-
tion which it is always communicating to the matter of the vw-
tex. Without fuch a principle it will undoubtedly cometo pals
Uiat the globe and the inward parts of the vortex, bdng al-
ways propagating their motion to the outward parts, and not
receiving any new motion, will gradually move flower and
flower, and at lafl, be carried round no longer.
Cob. 5. If another globe fliould be fwimming in the fame
vortex at a certain diftance from its centre, and in the meaa
time by fome force revolve conftantly about an axis of a
given inclination, the motion of this globe will drive the fluid
round after the manner of a vortex ; and at firft this new.
and fmall vortex will revolve with its globe about the Centre
of the other ; and in the mean time its motion will creep on
farther and farther, and by d^ees be propagated in vifini^
turn, after the manner of the firft vortex. And for the fame
reafon that the globe of the new vortex was carried about be-
fore by the motion of the other vortex, the globe of this
other will be carried about by the motion of this new vortex.
SeS. IX. OF NATURAL PHILOSOI^ItT. 153
lo that the two globes will revolve about fome intermediate
pointy and by reafon of that circular motion mutually fly
finom each other, unlefs fome force reftrains them. After*
warda^ if the conftantly imprefled forces, by which the globes
perfevere in their motions, fhould ceafe, and every thing be
left to aft according to the laws of mechanics, the motion of
the globes will languifh by degrees (for the reafon afligned in
€»r. 3 and 4), and the vortices at laft will quite ftand ftill.
CoR. 6. If feveral globes in given places fliould conftantly
revolve with determined velocities about axes given in pofition>
there would arife from them as many vortices going on in in*
fidiwn. For upon the fame account that any one globe pro*
pagates its motion in infinitum, each globe apart will propa-
gate its own motion in infinitum alfo ; fo that every part of
the infinite fluid will be agitated with a motion refulting from
the a6tions of all the globes. Therefore the vortices will not
"be confined by any certain limits, but by degrees run mutually
into each other ; and by the mutual adlions of the vortices on
each other, the globes will be perpetually moved from their
places, as was fhewn. in the laft corollary ; neither can they
poffibly keep any certain pofition among themfelves, unlefs
fome force reftrains them. But if tbofe forces, which are
oonflaQtly impreiTed upon the globes to continue thefe nK>-
tions, Ihould ceafe, the matter (for the reaibn aftigned in
cor. 3 and 4) will gradually ftop, and ceafe to move ia
vortices.
Cor. ?• If a fimilar fluid be inclofed in a fpherical veflel,
imdjby the uniform rotation of a globe in its centre, is driven
Toand in a vortex ; and the globe and vefl*el revolve the fame
way about the l^me axis, and their periodical times be as the
Iqnaies of the femi-diameters ; the parts of the fluid will not
go on in their motions without acceleration or retardation, till
their periodical times are as the fquares of their diftances
from the centre of the vortex. No conflitution of a vortex
can be permanent; but this.
CoR. >8. If the veflel, the inclofed fluid, and the globe,
retain this motion, and revolve befides with a common angu*
lar motion about any given axis, becaufe the mutual attri-
»^
154 MATHEMATICAL FtlNClPLBf, Book Q.
tion of the parts of the fluid is not climnged bj this motion,
the motions of the parts among each other will not be cbaiig*
ed ; for the tranflations of the parts among themfdves de*
pend upon this attrition. Any part will perfevere in that
motion in which its attrition on one -fide retards it jnft a*
Viuch as its attrition on the other fide accelerates it.
Cob. 9. Therefore if the veflfel be quiefcent, and the miK<
tion of the globe be given, tlie motion of the fluid will be
given* For conceive a plane to pais through the axis of the
globe, and to revolve with a contrary motion ; and fappofe
the fura of the time of this revolution and of the revdolioil
of the globe to be to the time of the revolution oi the gjobe
as the fquare of the femi-diameter of the vefiel to the ^naie
of the femi*diameter of the globe ; and the periodic times of
the parts of the fluid in refpedl of this plane will be as ibB
Squares of their difl;ances from the centre of the globe.
Cob. 10. Therefore if the veflel move about the fiune
with the globe, or with a given velocity about a different
the motion of the fluid will be given. For if from the whole
fyfliem we take away the angular motion of the veflTd^ a& the
motions will remain the fame among themfelves as befoire:,^lij^
cor. 8, and thofe motions will be given by cor. 9*
CoE. J 1 . If the vefTel and the fluid are quiefcent, and the
globe revolves with an uniform motion, that Inotion will be
propagated by degrees through the whole, fluid to thcTeflSd,
and the veflel will be carried round by it, unlefs violently de*
tained ; and the fluid and the veflel will be continually aoedte-
rated till their periodic times become equal to the . periodic
times of the globe. If the veflel be eitlier withheld by fome
force, or revolve with any confl;ant and uniform motiott^ the
medium will come by little and little to the fiate of motiea
defined in cor. 8, 9^ 10, nor will it ever perfevere in any odier
fiate. Bub if then the forces, by which th$ globe and vcffsl
revolve with certain motions, fliould ceafe, and the wh^
fyftem be left to a6l according to the mechanical laws, the
veflel and globe, by means of the intervening fluids will aft
upon each other, and will continue to propagate theif motions^
S€&. IX. OP KATUBAL PHILOSOPHY. 155
throagfa the fluid to each other^ till their periodic times be-
oome eqaal among themfelves^ and the whole fyfiem xeTolveft
together Uke one folid body.
SCHOLIUM.
In alLthefe reafooiinga 1 fuppofe the fluid to confift of mat-
ter gS nnifonn denfity and fluidity ; I mean, that the fioid is
fiichj that a globe placed any where therein may propagate
with the. fame motion of its own, at diftances from itfelf xod«^
turaally equal, fimilar and equal motions in the fluid in the
iame interval of time. The matter by its circular motion en-
dewoors to recede from the axis of the yortex> and therefore
piefles all the matter that lies beyond. This preflure makes
t|ie attrition greater, and the feparation of the parts more
difficult ; and by confequence diminiflies the fluidity of the'
matter. Again ; if the parts of the fluid are in any one place
deafer or larger than in the others, the fluidity will be kfs in
that place, becaufe there are fewer fuperficies where the parts
caa he feparated from each other. In thefe cafes I fuppofe
the defe& of the fluidity to be fupplied by the fmoothnefs. or
HoStisfify of the paufts, or fome other condition ; otherwife the
jpatter where it is lefs fluid will cohere more, and be more
ihiggifii, and therefore will receive the motion more flowly,
and propagate it farther than agrees with the ratio above
affigned. If tiie veflel be not fpherical, the particles will
mnve in lines not circular, buL anfwering to the figure of tlie
wflEbl ; and the periodic times will be nearly as the fquares of
the mean diftances from the centre. In die pai'ts between
the centre and the circumference the motions will be flower
where the fpaces are wide, and fwifter where narrow ; but
yet the particles will not tend to the circumference at all the
more for their greater fwiftnefs ; for they then defcribe arcs
of Ids cocvity, and the conatus of receding from the centre is
as. much diminiihed by the diminution of this curvature, as it
is augment&d by the increafe of the velocity. As they go out
of natrow into wide fpaces, they recede a little farther from the
oentiej but in doing fo are retarded ; and when they come
out of wide into narrow fpaces, they are again accelerated ;
and fo each particle is retaideid and accelerated by turns tor
)56 MATflEMATICAL PRINCIPLES 6ook II.
ever. Tliefe things will ;come to pafs in a rigid veflel ; for
the Hate of vortices* in an infinite fluid is known by cor. 6 of
this propofition.
I have endeavoured in this propofition to inveftigate the
properties of vortices^ that I might find whether theceleflaal
phsenbmena can be explained by them ; for the phsno-
menon is this^ that the periodic times of the planets revolving
about Jupiter are in the fefquiplicate ratio of their diftances
from Jupiter's centre ; and the fame rule obtains alfo among
the planets that revolve about the fun. And thefe rules ob-*
tain alfo with the greateft accuracy, as far as has been yet
^ifcoveredby aftronomical obfervation. Therefore if thofe
planets are carried round in vortices revolving about Jupiter
and the fun, the vortices muft revolve according to that law.
But here we found the periodic times of the parts of the vor-
tex to be in the duplicate ratio of the diftances from the cen^x
tre of motion ; and this ratio cannot be diminiihed and re*
duced to the fefquiplicate, unlefs either the matter of the vor-
tex be more fluid the farther it is from the centre, or the re*
fifl;ance arifing from the want of lubricity in the parts of the
fluid fliould, as the velocity with which the parts of the fluid
are feparated goes on increafing, be augumented with it in a
greater ratio than that in whidi the velocity increafes. Bat
neither of thefe fuppofitions feem reafonable. The m<H:e
grofs and lefs fluid parts will tend to the circumference, unkft
they are heavy towards the centre. And though, for the fake
of demonftration, I propofed, at the beginning of this fedtion,
an hypothefis that the refiftance is proportional to the veloci-
ty, neverthelels, it is in truth probable that the refiftance is in
a lefs ratio than that of the velocity ; which granted^ the
periodic times of the parts of the vcnrtex will be in a greater
than the duplicate ratio of the difl^mces from its centre. If,
as fome think, the vortices mov^ more fwiftly near the centre,
then flower to a certain limit, then again fwifter near the cir-
cumference, certainly neither the fefquiplicate, nor any other
certain and determinate ratio, can obtain in them. Let philo-
fophers then fee how that phenomenon of the fefquiplicate
ratio can be accounted for by vortices.
StS. XL OP NATURAL PHILOSOPHY. 157
PROPOSITION LIII. THEOREM XLI.
Bodies carried about in a vortex, and returning in the fame
orb, are of the fame denfity with the vortex, and are moved
according to the fame law with the parti of the vortex, as to
velocity and dire&ion of motion.
For if any fmall part of the vortex^ wliofe particles or pby-
fical points preferve a given fituation among each other^ be
fuppoled to be congealed^ this particle will move according
to the lame law as before^ fince no change is made either
in its denfity^ vi$ infita, or figure. And again ; if a congealed
or folid part of t;he vortex be of the fam:^ denfity with the reil
of the vortex^ and be refolved into a fluid, this will move ac-
cording to the fame law as before, except in fo far as its par-
tides, now become fluid, may be moved among themfelves. *
Negle^ therefore, the motion of the particles among them*
Iblves as not at all concerning the progreflive motion of the
whole, and the motion of the whole will be the fame as be-
fore. But this motion will be the fame with the motion of
other parts of the vortex at equal diflances from the centre;
becauie the folid, now refolved into a fluid, is become perfe&-
]j like to the other parts of the vortex. Therefore a folid,
if it be of the fame denfity with the matter of the vor-
tex, will move with the fame motion as the parts thereof,
being relatively at refl in the matter that furrounds it. If
it be more denfe, it will endeavour more than before to
recede from the centre; and therefore overcoming that
force of the vortex, by which, being, as it were, kept in
eqnilibrio, it was retained in its orbit, it will recede from
the centre, and in its revolution defcribe a fpiral, returning
no longer into the fame orbit. And, by the fame argu««
ment, if it be more rare, it will approach to the centre.
Therefore it can never continually go round in the fame or-
bit, nnleis it be of the fame denfity with the fluid. But
we have (hewn in that cafe that it would revolve accord-
ing to the fame law with thofe parts of the fluid that are
at the fame or equal diflances frou) the centre of the
vortex.
138 MATHEMATICAL PRIKCnPLlS BookU.
Cob. 1. Therefore a folid revolving in a vortex^ and. con-
tinually going ronnd in the fame orbit^ is relatively quiefcent
in die fluid that carries it.
Coa. 4. And if the vortex be of an nniform denfily^ the
fame body may revolve at any difiance from the centre of
tba vortex.
SCHOLIUM.
Hence it is manifeft that the planets are not carried round
in corporeal vortices ; for^ according to the Copemican hy*
pothefis^ the planets going round the fun reVolve in dlipfes,
having the fun in' their common focus; and by radii drawn to
the fun defcribe areas proportional to the tinies. But now
the parts of a vortex can never revolve with fucfa a motion.
I^et AD, BE, CF (PI. Q, Fig. 3), reprefent three orbits defciib«
ed about the fun S, of which let the utmoft cirdf; CF he
concentric to the fun; and let th^ aphelia of the two inner-
moft be A, B; and their perihelia D, E. Therefore' k body
revolving in the orb CF, defcribing, by a radius drawn to
the fun, areas proportional to the times, will move with an
uniform motion. And, according to the laws of aftronomy^
the body revolving in the orb BE will move flower in its aphe<p
lion B, and fwiflter in its perihelion £; whereas, according
to the laws of mechanics, the matter of the vortex ought
to move more fwiftly in the narrow fpace between A and C
than in the wide fpace between D and F; that is, more
fwiftly in the aphelion than in the perihelion. Now thefe
two conclufions contradid): each other. So at the beginning
of the fign of Virgo, where the aphelion of Mars is at, pre-
fent, the diftance between the orbits of Mars and Venus is to
the diftance between the fame orbits, at the beginning of the
fign of Pifces, as about 3 to 2 ; and therefore the matter of
the vortex between thofe orbits ought to be fwifter at the be-
ginning^of Pifces than at the beginiung of Virgo in the ratio
of S to 2 ; for the narrower the fpace is through Which the
fame quantity of matter pafTes in the fame time of one revo-
lution, the greater will be the velocity with which it paflJes
through it. Therefore if the earth being relatively at rfeft in
this celeftial matter (hould be carried round by it, and revolve
.^.J-
VlaUXJoin.
/la^cJSZ
I
I I •
I I I
r.li.ii
n
Sl.r
1 1 1
A B;C:i>-R
XT
b
ii:
iillii:!
HXtm/iuut, Set/ Wufrf^Um Str^^^A'^hr/v
SkS. IX. OF NATURAL PHILOSOPHY. 159
together with it about the fun^ the velocity of the earth at the
bc^nning of Pifces would be to its velocity at the beginning
of Virgo in a fefquialteral ratio. Therefore the fun's apparent
diamal motion at the beginning of Virgo ought to be above
70 minutes^ and at the beginning of Pifces lefathan 48 mi-
nutes; whereas^ on the contrary^ that apparent motion of
the fun is really greater at the beginning of Pifces than at
the beginning of Virgo, as experience teftifies ; and therefore
the earth is fwifter at the beginning of Virgo than at the be-
ginning of Pifces : fo that the hypothefis of vortices is utter-
ly irrecohcileable with aftronomical phaenomena^ and rather
ferves to perplex than explain the heavenly motions. How
tbefe motions are performed in free fpace$ without vortices,
may be underftood by the firft book ; and I fhall now mote
fully treat of it in the following book.
BOOK III.
IN the preceding books I have laid down the principles
ofphilofophy; principles not philofophical, but mathematical;
fuchi to Wit, as we may build our reafoniugs upon in philo«
fophical enquiries. Thefe principles are the laws and condi-
tions of certain motions, and powers or forces, which chiefly
have refpe£b to philofophy ; but, left they fhould have appear-
ed of themfelves dry and barren, I have illuftrated them here
and there with fome philofophical fcholiums, giving an ac-j
count of fuch things as are of more general nature, and which
philofophy feems chiefly to be founded on ; fuch as the den-
fity and the refiftahce of bodies, fpaces void of all bodiesj,
and the motion of light and founds. It remains that, from
the fame principles, I now demonftrate the frame of the
Syftem of the VTorld. Upon this fubjeft I had, indeed, com-i
pofed the third book in a popular method, that it might be
read by many ; but afterwards, confidering that fuch as had
not fufficiently entered into the principles could not ealily
difcem the flrength of the confequences, nor lay aiide the
prejudices to which they had been many years accuftomed,
therefoxe^ to prevent the difputes which might be raifed upo«
IGO MATHEMATICAL FBINCfPLBS Book III.
fucb accounts^ I cbofe to reduce the fubftance of this book
into tbe form of propofitions (in the matbematical wajr)^
which (hould be read by tbofe only who had firft made them-
felves mafters of the principles eftabliflied in the preceding
books: not that I would advife any one to the previous fiudj
of every proportion of tbofe books; for they abound with
fucb as might coft too much time^ even to readers of good
mathematical learning. It is enough if one carefully reads
the definitions^ the laws of motion^ and the iirft three feAions
of the firft book. He may then pafs on to this book^ and
confult fuch of the remaining propofitions of the fiift two
books^ as the references in this^ and bis occafions^ fliall
require.
RULES OF REASONING IN PHILOSOPHY.
RULE I.
JVe are to admit no more caufes of natural thvigs than fuch
as are both true andfufficient to explain their appearances*
To this > purpofe the philofophers fay that Nature does
nothing in vain^ and more is in yain when lefs will ferve ;
for Nature is pleafed with fimplicity^ and affedls not the pomp
of fuperfiuous caufes.
RULE II.
Therefore to the fame natural effe£ts we mujl, as far aspojfible,
affign the fame caufes.
As to refpiration in a man and in a beaft ; the defceni of
ftones in Europe and in America ; the light of our culinary
fire and of the fun ; the refledlion of light in the; earthy and in
the planets.
RULE in.
The qualities of bodies, which admit neither intension nor re-
mifjion of degrees, and which are found to belong to all bo"
dies within the reach of our experiments, are to be eftumed
the univerfal qualities of all bodies whatfoever.
For fince the qualities of bodies are only known to us by
experiments^ we are to hold for univerfal all fuch as univer^-
fally agree with experiments ; and fuch as are not liable to di-
minution can never be quite taken away. We are certainly
Book IIL OF NATURAL PHILOSOPHY. l6l
not to relinquifh the evidence of experimeDts for the fake of
dreams and vain fiAions of our own deviiing ; nor are we to
recede from the analogy of Nature^ which ufes to he fimple,
and always coofonant to itfelf. We no other way know the
extenfioH of bodies than by our fenfes^ nor do thefe reach it
in -all bodies ; but becaufe we perceive extenfion in all that
are fenfible^ therefore we afcribe it univerfally to 9II others
alfo. That abundance of bodies are bard^ we learn by expe^
rience ; and becaufe the hardnefe of the whole arlfes from the
hardnefs of the parts^ we therefore juilly infer the hardnefs of
the undivided particles not only of the bodies we feel but of
all others. That all bodies are impenetrable^ we gather not
from reafon^ but from fenfation. The bodies which we handle
we find impenetrable^ and thence conclude impenetrability to
be an univerfal property of all bodies whatfoever. That all
bodies are moveable^ and endowed with certain powers (which
we call the vires inertia) of perfevering in their motion^ or in
their reft, we qply infer from the like properties obferved in
the bodies which we have feen. The extenfion, hardnefs.
Impenetrability, mobility, and vis inertia of the whole, refult
from the extenfion, hardnefs, impeuetrability, mobility, and
vires inertia of the parts ; and thence we conclude the leaft
particles of all bodies to be alfo all extended, and hard, and
impenetrable, and moveable, and endowed with their proper
vires inertia. And this is the foundation of all pbilofophy.
Moreover, that the divided but contiguous particles of bodies
may be feparated from one another, is matter of obfervation ;
and, in the particles that remain undivided, our mii^ds are
able to diftinguifh yet lefTer parts, as is mathematically de-
monflrated. But whether the parts fo difiinguifhed, aiid not
yet divided, may, by the powers of Nature, be adlually di-
vided and feparated from one another, we cannot certainly
determine. Yet, had we the proof of but one experiment that
any undivided particle, in breaking a hard and folid body,
fu£fered ^ divifion, we might by virtue of this rule conclude
that the undivided as well as the divided particles may be di-
vided and a^ually feparated to infinity*
Laftly, if it univerfally wpears, by experiments and aftro-
nomical obfervatiops^ that all bodies about the earth grayir
Vol, II, M
IGS . MATHEMATICAL PftlMClPLBS Book III.
tate towards the eaitb, and that in proportioo to thequanti^
of matter which they fererally contain ; that the moon like*
wife^ according to the quantity of its matter, gratitates to-
wards the earth ; that, on the other hand, oar fea gravitates to*
wards the moon ; and all the planets mutually one towards
another ;^and the comets in like manner towards the fun ; we
mnft^ in confequence of this rule, univerfally allow that all
bodies whatfoever are endowed with a principle of mutual gra^
Titation. For the argument from the appearances concludes
with more force for the univerfal gravitation of all bodies
than for their impenetrability ; of which, among thofe in the
celeftial regions, we have no experiments, nor any manner
of obfervation. Not that I affirm gravity to be eflential to
bodies : by their vis infita I mean nothing but their m ta-
ertidi. This is immutable. Their gravity is diminilhed as they
recede from the earth.
RULE IV.
In experimental philofophy tee are to look upon propqfitiom
colleSed by general induSionfrompfutnomenaai acatrately
or very nearly true, notwithfianding any contrary hypathefa
that may be imagined, tillfuch time as other phenomena
occur, by which they may either be made more accurate, or
liable to exceptions.
This rule we muft follow^ that the argument of induAion may
not be evaded by hypothefes.
PILENOMENA, OR APPEARANCES.
PH/ENOMENON I.
Huit the circunyovial planets, by radii drawn to Jupiteir^s cen*
tre, defcribe areas proportional to the times of defcription ;
and that their periodic times, the fixed Jiars being at refi,
are in the fefquiplicate proportion of their diftances from
its centre.
This we know from aftronomical obfervations. For the or-
bits of thefe planets differ but infenfibly from circles concen*
tiic to Jupiter; and their motions in thofe circles are found
to be uniform. And all afhronomers agree that their periodic
times are in the fefquiplicate proportion of the fe]Bi-diameter»
of their orbits ; and fo it manifeftly af^ars from the fbllow-
iog table
I
Bdok UL t>¥ NATURAL PHILOSOPHY* \QS
lie periodic times of the fatelliies of JupittTp
. 1^.18^,27' 34''. 3*. 1S\ 13' 42"- 7V3\ 4£' S6".
16^ 16\ 32' , 9".
Tke diftance^ of ihefatelRtesfrom Jupiteft centre.
Borelli .—
tOfmAMj by Hi§ Ukt9wu — ..
Q.dSktXhmikgTltlrfmpe, -
Caffini by tkt eeUp, cfihefaieU
WhfmiheSerMkHmei mm^m»»~
1
H
5
H
25355 '
Mr. Poibullias detennined^ by the help of excellent micro-
meters; the diameters of Jupiter and the elongation of its fa-
teliitea after the following manner. The greateft heliocentric
elongation' of the fonrtli fatellite from Jupiter's centre was
taken with a micrometer in a 15 feet telefcope, and at the
mean difiance of Jupiter from the earth was found about
a' 16". : Th^ elongation of the third fatellite was takeh with
a micrometer in a telefcope of 123 feet^ and at the fame dift-
ance of Jupiter from the earth was found 4' 42". The greateft
elongations of the other fatellites^ at the fame diftance of Ju-
piter from th6 earthy are found from the periodic times to be
S'56"47'", andl'51" 6"'.
The diameter of Jupiter taken with the micrometer in a
123 feet telefcope feveral times^ and- reduced to Jupiter's
mean diftance from the earthy proved always lefs than AQt',
never lefs than 38", generally S^'. This diameter in ihorter
telefcopes is -40", or 41" ; for Jupiter's light is a little dilated
by the unequal refrangibillty of the rays, and this dilatation
bears a lefs ratio to the diameter of Jupiter in the longer and
more perfed): telefcopes than in thofe which are ihorter and
lc&perfe&. The limes in which two fatellites, the firft and
the third, paffed over Jupiter's body, were obferved, from the
beginning of the ingrefa to the beginning of the egrefs, and
from the complete ingrefs to the complete egref6> with the
long telefcope. And from the tranfit of the firft fatellite, the
diameter of Jnpiter at its mean diftance from the earth came
*Mrth S7V'> «fld from the traofit of the third S7|". There
was obferved alfo the time in which tlie fhadow of the firft
fatellite paifled over Jupiter's body, and thence the diameter
of Joplter at its mean diftance from the earth caine out about
M 2
l64 11 ATHEMATICAL FftlKClPLES Book lU.
37''. Let 118 foppofe its diameter to be 37 i" verf neaily, and
then the greateft dongations of the firft^ fecond^ tbird^ and
fourth fatellite will be refpeAivelj equal to 5^96^^ 9>494>
15,141, and£6j63 femi-diameiers of Jopiter.
PHiENOMENON 11.
That the eireumfatumal planets, by radii drawm to Satum^s
centre, deferibe areas proportional to the times ofdefcr^
tion ; and that their periodic times, the fixed jtars bemg at
refi, are in the fejquipUcaie proportion of their di^anees
from its centre.
For^ as Caffini from his own obfervationa has determined,
their diftances from Saturn's centre and their periodic times
are as follow.
The periodic times of thefatellites of Saturn.
1*. ^IK 18' 27". 2^. 17^ 4V2&". 4^. l^. ^^' 19^. 15*- £2^.
41' 14''. 79^. 7^. 48' 00".
l%e difiances of thefatellites from Saturn's centre, inftnu-duH
meters of its ring.
From obferoatiom • \^. 2{ 3f. 8. 24.
From the periodic times 1^93. 2^47- 3,45. 8. 23,35.
The greateft elongation of the fourth fatellite from Saturn's
centre is commonly determined from the obfervations to be.
eight of thofe femi-diameiers very nearly. But the greateft
elongation of this fatellite from Saturn's centre, when taken
with an excellent micrometer in Mr. Hwfgens's telefcope pf
123 feet, appeared to be eight femi-diameters and i^ of a
femi-diameter. And from this obfervation and the periodic
times the diftances of the fatellites from Saturn's centre ia
femi-diameter$ of the ring are 2,1, 2,69. 3,75. 8,7. und 25,35.
The diameter of Saturn obferved in the fame telefcope was
found to be to the diameter of the ring as S to 7 ; and the
diameter of the ring. May 28-29, 17 19> w^a found to be 43'' ;
and thence the diameter of the ring whf^n Saturn is at its
mean difiance from the earth is 42", and the diameter of Sa-r
turn 18". Thefe things appear fo in very long ^d excellent
t^efcopes, becaufe in fuch telefcopes the apparent magnir
tudes of the heavenly bodies bear a greater proportioa to the
dilatation of light m the extx^ities of tbo^ boclie^ tha^/iA
JSook IIL OF NATUHAL PHILOSOPttV^ 165
Ihorter tekfoopes. If we^ then^ rejedl all the fpurious lights
the diameter of Saturn will Dot amount to more than 16".
PHiENOMENON III.
That thejive primary planets, Mercury, Fenus, Mars, Jupiter,
and Saturn, mth their feveral orbits, encompafs the fun.
That Mercury and Venus revolve about the fun^ is evident
from their moon-like appearances. When they fhine out
with a full face^ they are^ in refpe6); of us^ beyond or above
the fun; when they appear half full^ they are about the fame
hei^tonone fide or other of the fun; when horned^ they
are below or between us and the fun ; and they are fometimes^
when direSly under, feen like fpots traverfing the fun's diik«
That Mars furrounds the fun^ is as plain from its full face
when near its conjundtion with the fun^ and from the gibbous
figure which it Ihews in its quadratures. And the fame
thing is demonftrable of Jupiter and Saturn^ from their ap-
pearing full in all fituations; for the fhadows of their fatellites
that appear fometimes upon their difks make it plain that
the light they fliine with is not their own^ but borrowed
from the fun.
PHiENOMENON IV.
That the fixed Jiari being at reft, the periodic times of the five
primary planets, and (whether of the fun about the earth,
or) of the earth about the fun, are in the fefquiplicate pro'
portion of their mean diftancesfrom the fun.
This proportion^ firfl. obferved by Kepler, is now received
by all aflronomers; for the periodic times are the fame^
and the dimenfions of the orbits are the fame^ whether the
fxaoi revolves about the earthy or the earth about the fun.
And as to the meafures of the periodic times^ all aflronomers
are agreed about them. But for the dimenfions of the orbits^
Kepler and Bullialdus, above all others^ have determined
tfaem from obfervations with the greatefl accuracy; and the
Qiean diftances correfpondiug to the periodic times differ
Imt infenfibly from thofe which they have afTigned^ and for
the moft part fall in between them; as we may fee from the
fallowing table. • M 3
]66 BfitnsMATioAL niNoiPLBt Sboi III.
ITier periodic times, with refpeSt to the fixed Jlar$, of the p&i-
neti and earth revolviog about the fun, in day$ and deei"
mal parts of a day.
10759^275. 4339,514. 686,9785. 365,2565. £M,6l76.
»
87,9692.
The mean difiances of the planets and of the earth from the
fun.
h % ' *
According to Keplef .95 1000. 5 19650. 152350.
to BuUialdus 954198. 522520. 152350.
to the periodic times . 954006. 520096. 1 52369.
8 ? 9
According to Kepler lOOOOO. 72400. 38806. *
ioBullialdus lOOOOp. 72398. 38585.
to the periodic times . .100000. 72333. 38710.
As to Mercury and Venus, there can be no doubt about
their difiances from the fun; for they are determined by the
elongations of thofe planets from the fun; and for- the
difiances of the fuperior planets, all difpute is cut off by the
eclipfes of the fatellites of Jupiter. For by thofe eblipfes
the pofition of the fhadow which Jupiter projeAsis deter-
mined; whence we have the heliocentric longitude of Jupi-
ter. And from its. heliocentric and geocentric longitudes
compared together, we determine its diftance.
PHENOMENON V.
Then the primary planets, by radii drawn to the earth, defcribe
areas no wife proportional to the times ; but that the
areas which the^y defcribe by radii drawn to the fun are
proportional io the times of defcription.
For to the earth they appear fometimes direcSl, fometmies
ftationary, nay, and fometimes retrograde. But finom the
fun ttey are always feen dire<S, and to proceed with a tnotion:
nearly uniform, that is to fay, a little fwifter in the perihelion
and a little (lower in the aphelion difiances, fo as to main-
tain an equality in the defcription of the areas. This a noted
propofition among aftronomers^ and particularly demonfira-
Book in. or NATUBAL PHILOSOPHY. 167
ble in Jupiter^ from the eclipfes of his fatellites; by the help
of which eclipfes^ as we have faid^ the heliocentric longitudes
of that planet^ and its diftances from the fun^ are determined,
PfliENOMENON VI.
That the moon^ by a radius drawn to the eartKs centre,
defcribes an area proportional to the time of defcription.
This we gather from the apparent motion of the moon^ com-i
pared with its apparent diameter. It is true that the motion
of the moon is a little difturbed by the a6);ion of the fun:
but in laying down thefe phsenomena, I negle6l thofe fmall
and inconfiderable errors.
PROPOSITIONS.
PROPOSITION I. THEOREM I.
l%at the forces by which the circumjovial planets are conti"
nually drawn off from reBilinear motions, and retained in
their proper orbits, tend to Jupiter^s centre; and are rc-
ciprocally as the fquares of the diftances of the places of
thofe planets /ram that centre.
The former part of this propofition appears from phasn. ]^
and prop, 2 or S, book 1 ; the latter from phaen. 1, and cor.
6, prop. 4i of the fame book.
The fame thing we are to underftand of the pljinets which
encompafs Saturn^ by phsen. ^.
PROPOSITION IL THEOREM II.
.That the forces by which the primary planets are continually
drawn off from re&ilinear motions, and retained in their
proper orbits, tend to the fun; and are reciprocally as the
fquares of the diftances of the places of thofe planets ^om
thejun^s centre.
The former part of the propofition is manifefl from phsen.
5, and prop. 2, book 1 ; the latter from phaen. 4, and cor. 6,
prop. 4, of ^he fame book. But this part of the propofi-
don is, with great accuracy, demonftrable from the quiefcence
€^ the aphelioft points ; for a very fmall aberration from the
reciprocal duplicate proportion would (by cor. 1, prop. 45,
book ]) produce a motion of the apfides fenfible enough in
every fingle revolution, and in many of them enormoufly
great. M 4
168 MATHEMATICAL PBINCIPLES BookllL.
PROPOSITION III. THEOREM III.
That the force by which the moon is retained in its orbit tends
to the earth ; and is reciprocally as thefquare ofthtdiftance
of its place from the earth's centre.
The former part of the propoiition is evident from phsn. 6^
and prop. 12 or 3^ book 1 ; the latter from the very flow mo-
tion of the moon's apogee ; which in every Angle revotation
amounting but to 3® 3' in confequentia, may be negleiSted.
For (by cor. 1^ prop. 45, book 1) it appears, that, if the
diilance of the moon from the earth's centre is to the femi-
diameter of the earth as D to 1^ the force^ from which fuch
a motion will refult^ is reciprocally as D^^J-j, i. e. reciprocally
as the power of D^ whofe exponent is £777 ; that is to fay^ in
the proportion of the diflance fomething greater than reci-
procally duplicate^ but which comes 59i times nearer to the
duplicate than to the triplicate proportion. But in regard
that this motion is owing to the adlion of the fun (as we (hall
afterwards fhew)^ it is here to be negle6led. The a^lion of the
fun, attraaing the moon from the earth, is nearly as the
moon's diftance from the earth ; and therefore (by what we
have fhewed in cor. 2, prop. 45, book 1) is to the Centripetal
force of the moon as 2 to 357,45, or nearly fo ; that is, as 1
to 178H* And if we negled); fo inconfiderable a force of tihe
fun, the remaining force, by which the moon is retained in
its orb, will be reciprocally as D*. This will yet more fiilly
appear from comparing this force with the force of gravity, as
is done in the next propofition.
CoH. If we augment the mean centripetal force by whica
the moon is retained in its orb, firH in the proportion of
mH to 178Hi and then in the duplicate propcnrtion of the
femi-diameter of th^ earth to the mean diftance of tbe cen-
tres of the moon and earth, we Ihall have the centripetal
force of the moon at the furface of the earth ; flippofing this
force, in defcending to the earth's furface, continually to
increafe in the reciprocal duplicate proportion of the
height.
Book ILL OF NATURAL PHILOSOPHY. l6g
PROPOSITION IV. THEOREM IV.
That the moon gravitates towards the earth, and by the force
of gravity js continually drawn off from a re&ilinear motion,
and retained in its orbit.
The mean diftance of the moon from the earth in the fyzy-
gies in femi-diameters of the eaith^ is^ according to Ptolemy
and moft ailronomers^ 59 ; according to Vendelin and Huy'-
gens, 60; to Copernicus, 60^; to Street, 60^ ; and to Tycho, 56|.
But Tycho, and all that follow his tables of refradion^ mak-
ing the refra£lions of the fun and moon (altogether againft
the nature of light) to exceed the refradlions of the fixed ftars^
and that by four or five minutes near the horizon, did thereby
increafe the moon's Aorfzoz/^aZ parallax by a like number of
minutes^ thatis^ by a twelfth or fifteenth part of the whole
parallax. Correct this error^ and the diftance will become
about 60| femi-diameters of the earthy near to what othgrs .^i^/v
have afiSgned. Let us aiTume the mean diftance of^ 60 dia-
meters in the fyzygies ; and fuppofe one revolution of the
moon^ in refpeft of the fixed ftars, to be completed in 27^. 7*»,
43^ as aftronomers have determined ; and the circumference
of the earth to amount to 1^3249600 Paris feet^ as the French
have found by menfuration. And now if we imagine the
moon^ deprived of. all motion^ to be let go^ fo as to defcend
towards the earth with the impulfe of all that force by which
(by cor. prop. 3) it is retained in its orb, it will, in the fpace
of one minute of time, defcribe in its fall 15^ Paris feet.
This we gather by a calculus, founded either upon prop. S6,
book 1, or (which comes to the fame thing) upon cor. 9, prop.
4b of the fame book. | For the verfed fine of that arc, which
the moon^ in the fpace of one minute of time, would by its
mean motion defcribe at the diftance of SO femi-diameters of
the. earthj is nearly 15^ Paris feet, or more accurately \C
feetj 1 inch, and 1 line J. Wherefore, fince that force, in
approaching to the earth, increafes in the reciprocal dupli*
cate proportion of the diftance, and, upon that account, at
the furface of the earth, is 60 x 60 times greater than at
the mbon^ a body in our regions, falling with that force,
ougljit^ in the fpace of ooe. minute of time^ to defcrU)e
170 MATHEMATICAL PRlKCIPLEif Book HI. ^
Go X 60 X Id-h Ports feet; and^in the fpaceof one fecond
of time, to defcribe 15^7 of thofe feet ; or more accurately
15 feet, 1 inch, and 1 line ^ And with this very force we
adlually find that bodies here upon earth do really defcend ;
for a pendulam ofcillating feconds in the latitude of Parts
wfli be 3 Paris feet, and 8 lines i in length, as Mr. Huygcns
hasobferved. And the fpace which a heavy body defcribes
by falling in one fecond of time is to half the length of this
pendulum in the duplicate ratio of the circumference of a
circle to its diameter (as Mr. Huj/gens has alfo (hewn), and is
therefore 15 Paris feet, 1 inch, 1 line -J.. And therefore the
force by which the moon is retained in its orbit becomes, at
the very furface of the earth, equal to the force of gravity
which we obferve in heavy bodies there. And therefore (Igr
rule 1 and i) the force by which the moon is retainedin its
orbit is that >ery fame force which we commonly call gra-
vity; for, were gravity another force different from that^
then bodies defcending to the earth with the joint impulfe of
both forces would fall with a double velocity, and in the fpace
^f one fecond of time would defcribe S0{- Paris feet ; alto*
gether againft experience.
This calculus is founded on the hypothefis of the earth's
fianding ftill ; for if both earth and moon move about the
fun, and at the fame time about their common centre of gra-
vity, the diftance of the centres of the moon and earth from
one another will be 60| femi-diameters of the earth ; as may
be foulad by a computation fromi prop. 60, book !•
SCHOLIUM.
The demonftration of this propofitibn may be more diffiifely
explained after the following manner. Suppofe feveral moons
to revolve about the earth, as in the fyftem of Jupiter or Sa-
turn ; the periodic times of thefe moons (by the argument of
indu<^on) would obferve the fame law which Kq)lcr found
to obtain among the planets ; and therefore their centripetal '
forces would be reciprocally as the fquares of the diftances
from the centre of the earth, by prop. 1, of this book. Now
if the lowefi of thefe were very fhiall, and were fo nefurthe
earth as almoft to touch the tops gf the highefi mouHtaios,
BoBk in. O^ NATURAL PHILOSOPHY. 171
the centripetal force thereof^ retaining it in its.orbj would be
v^ nearly equal to the weights of any terrtjlrial bodies that
flionld be found upon the tops of thofe mountains^ as may be
known by the foregoing computation. Therefore if the fame
)i^tle moon (hould be deferted by its centrifugal force that
carries it through its orb^ and fo be difabled from going on-
wards therein, it would defcend to the earth ; and that with
the fame velocity as heavy bodies do adlually fall with upoa
the tops of thofe very mountains ; becaufe of the equality of
the forces that oblige them both to defcend. And if die
force by which that loweft moon would defcend were diflfer-
ent from gravity, and if that moon were to gravitate towards
the earth, as we find terreftrial bodies do upon the tops of
mountains, it would then defcend with twice the velocity, as
being impelled by both thefe forces confpiring together.
Therefore fince both thefe forces, that is, the gravity of heavy
faodies> and the centripetal forces of the moons, refped): the
centre of the earth, and are fimilar and equal between them-
felves, they will (by rule 1 and 2) have one and the fame
caufe« And therefore the force which retains the moon' jn
its orbit is that very force which we commonly call gravity;*
becaufe otherwife this little moon at the top of a mountain
mnft either be without gravity, or fall twice as fwifUy as heavy
bodies ufe to do.
PROPOSITION V. THEOREM V.
That the cireumjovial planets gravitate towctrds Jupiter; tht
drcumfatumal towards Saturn ; the circumfolar torvardi
the fun; and by the forces of their gravity are drawn off
from rtSilinear motions, and retained in curvilinear or-
bits.
' Tot the revolutions of the cireumjovial planets about Ju-
piter, of the circumfatumal about Saturn, and of Mercury
aiM' Venus, and the other circumfolar planets, about the fun,
are ap^^earafices of the fame fort with the revolution of the
moon aboat the earth ; and therefore, by rule 2, mufi be owing
to the faftbe fort of caufes ; efpecially fince it has been de^
lAoaftrated/ that the forces upon which thofe revolutions de-
pend tend to the centres of Jupiter, .pf Saturn^ and of the
172 MATHEMATICAL PRINCIPLES Book IIL
fan ; and that thofe forces^ in receding from Jupiter^ froo^
Saturn^ and from the fun^ 'decreafe in the fame proportion^
and according to the fame lawj as the force of gravity does
in receding from the earth.
Cor. 1. There is, therefore, a power of gravity tending to
all the planets ; for, doubtlefs, Venus, Mercury, and the reft^
are bodies of the fame fort with Jupiter and Saturn. And
fince all attradlion (by law 3) is mutual, Jupiter will therefore
gravitate towards all his own fateUites, Saturn towards bis^
the earth towards the moon, and the fun towards all the pri*
mary planets.
Cor. 2. The force of gravity which tends to any one
planet is reciprocally as the fquare of the diftance of placet^
from that planet's centre.
Cor. 3. All the planets do mutually gravitate towards one
another, by cor. 1 and 2. And hence it is that Jupiter and
Saturn, when near their conjundlion, by their mutual at-
tractions fenfibly difturb each other's motions. So the fun
difto^bs the motions of the moon ; and both fun and moon
difturb our ilea, as we fhall hereafter explain.
SCHOLIUM.
The force which retains the celeftial bodies in their orbits
has been hitherto called centripetal force ; but it being now
made plain that it can be no other thau a gravitating force,
we (hall hereafter call it gravity. For the caufe of that cen-
tripetal force which retains the moon in its orbit will extend
itfelf to all the planets, by nile 1, 2, and 4.
PROPOSITION VI: THEOREM VI.
That all bodies gravitate towards every planet ; and that the
weights of bodies towards arty • the fame planet^ at equal
difiances from the centre of the planet y are proportional to
the quantities of matter which they feverally contain*
It has been, now of a long time, obferved by others, that
all forts of heavy bodies (allowance being made for the in-
equality of retardation which they fuflFer from a fmall power
of reMance in the air) defcend to the edxihfrom equal he^hts
in equal times ; and that equality of times we may diflinguiih
to & great accuracy, by the help of pendulums. I tried the
Book in. OF NATURAL PHILOSOPHY. . 173
diing ill gold^ fHvety lead^ glafs^ fand^ common falt^ wood^
WBtor, aud wheat. I provided two wooden boxes^ round and
equal : I filled the one with wood^ and fufpended an equal
weight of gold (as exa^lly as I could) in the centre of ofcilla^
tion of the other. The boxes hanging by equal threads of 1 1
feet made a couple of pendulums perfe&ly equal in weight
and figure^ and equally receiving the refiftance of the air.
And^ placing the one by the other^ I obferved them to play
togethier forwards and backwards^ for a long time^ with equal
vibrations. And therefore the quantity of matter in the gold
(by cor« 1 and 6, prop. Q4, book Q) was to the quantity of
matter in the wood as the adlion of the motive force (or vis
motrix) upon all the gold to the a6lion of the fame upon all
the wood ; that is^ as the weight of the one to the weight of
the other : and the like happened in the other bodies; By
thefe experiments^ in bodies of the fame weighty I could
manifeftly have difcovered a difference of matter lefs than
the thoufandth part of the whole^ had any fuch been. But,
without all doubt^ the nature of gravity towards the planebi
is the fame as towards the earth. For^ ihould we imagine
our terreflxial bodies renioved to the orb of the moon^ and
there^ together with the moon^ deprived of all motion^ to be
let go^ fo as to fall together towards the earthy it is certain^
from what we have demonflrated before^ that^ in equal tinies,
they would defcribe equal fpaces with the moon^ and of con-
fequen,oe are to the mooi^ in quantity of matter^ as their
weights to its weight. Moreover^ iince the fatellites of Jupi-
ter perform their revolutions in times which obferve the fef-
qniplicate proportion of their diftances from Jupiter's centre^
their accelerative gravities towards Jupiter will be reciprocally
as the fquares of their diilances from Jupiter's centre ; that is^
equals at equal diftances. -And^ therefore^ thefe fatellites^ if
fuppofed to fall towards Jupiter from equal heights^ would
defcribe equal fpaces in equal times^ in like manner as heavy
bodies do on our earth* And^ by the fame argument^ if the
circumfolar planets were fnppofed to be let fall at equal dif*
lances from the fun^ they would, in their defcent towards the
fim, defqibe eqpal fpsces in e^jaeX times. But forces wl)ich
174 MATHEMATICAL PBINCIPLES > Book lit.
etjfMily liccelerate anecfual bodies muft be as tbofe bodies;,
tbat is to fay> the weights of the planets towardi iheftm moft
be as their qHantities of matter. Further^ that the weights
of Jupiter and of his fatellites towards the fun are proportional
to the fevenil quantities of their matter^ appears firom the ex«^
oeedingly regular motions of the fatellites (by cor. S^ {nrop. 65,
book 1). For if Ibme of thofe bodies were more ftrongly at*
traded to the fan in proportion to their quantity of maltet
than others^ the motions of the fatellites would be diftnrbed
by that inequality of attra6Uon (by cor, 2, prop. 65, book 1).
If, at equal diftances from the fun^ any fatellite^ in pippor*
tion to the quantity of its matter^ did gravitate towards the
fun with a force greater than Jtipiter in proportion to bis^ ac«
cording to any given proportion^ fuppofe of d to e ; then the
diftance between the centres of the fun and of the fatellite's
cfarbit would be always greater than the diftance between the
centres of the fun and of Jupiter nearly in the fubduplicate of
that proportion; as by fome computations I have found.
And if the fatellite did gravitate towards the fun with A force,
leiTer in the proportion of e to d^ the diftance of the centre
of the fatellite's orb from the fun would be lefs than the dif-^
tance of the centre of Jupiter from the fun in the fubduplicate
of the fame proportion. Therefore if, at equal diftances from
the fun, the accelerative gravity of any fatellite towards the fun
were greater or lefs than the accelerative gravity of Jupiter to-
wards the fun but by one -nnnr P^^^t of the whole gravity,
the diftance of the centre of the fatellite's orbit from the fun
would be greater or lefs than the diftance of Jupiter from the
fbn by one ^^^ part of the whole diftance ; that is, by a fifth
part of the diftance of the utmoft fatellite from the centre of
Jupiter ; an eccentricity of the orbit which would be very
feniible. But the orbits of the fatellites are concentric to
Jupiter, and therefore the accelerative gravities df Jupiter,
and of all its fatellites towards the fun, are equal among them-
felves. And by the fame argument, the weights of Saturn
and of his fatellites towards the fun, at equal diftances from
the fun, are as their feveral quantities of matter ; and the
weights of the moon and of the earth towards the fun are
Book UI. OP NATURAL PHILOSOPHY. ' 175
either aone, or accurately proportional to the mafles of mat*
ter which they contain. Bat fome they are, by cor. ] and 3,
prop. 5. .
Bui further; the weights of all the parts of every planet to^
wards any other planet are one to another as the matter in
the feveral parts; for if fome parts did gravitate more^
others kls, than for the quantity of t^eir matter, then- th^
whole planet, according to the fort of parts with which it
moft abounds, would gravitate more or lefs than in propor*-
tion to the quantity of matter in the whole. Nor is it of any
moment whether thefe parts are external or internal ; fot
if, for example, we fhould imagine the terreftrial bodies with
US to be raifed up to the orb of the moon, lo be there com-*
pared with its body ; if the weights of fuch bodies were to the
weights of the ext^nal parts of the moon as the quantities of
matter in the one and in the other refpedively ; but to the
weights of the internal parts in a greater or lefs proportion,
then likewife the weights of thofe bodies would be to tho
weight of the whole moon in a greater or lefs proportion ;
againft what we have (hewed above. -
Cor. 1. Hence the weights of bodies do not depend upon
their forms and textures ; for if the weights could, be altered
with the forms, they would be greater or lefs, according to the
Tarietyof forms, in equal matterjaltogetheragainftexperience.
CoR. d. Univerfally, all bodies about the earth gravitate
towards the earth ; and the weights of all, at equal diftances
from the earth's centre, are as the quantities of matter which
they feverally contain* This is the quality of all bodies
within the reach of our experiments ; and therefore (by rule
d) to be afiirmed of all bodies whatfoever. If the atherf or
any other body, were either altogether void of gravity, or were
to gravitate lefs in proportion to its quantity of matter, then,
becanfe (according to Arijlotle, Dcs Carter, and others) there
is no difference' betwixt that and other bodies but in mere
form of matter, by a fucoeffive change from form to form, it
might he changed at kit into a body of the fame condition
with thofe which gravitate moft in proportion to their quan-
ti^ of matter ; and^ on the other hand, the heavieft bodies^
176 MATHEMATICAL PBTNCTPLE9 Sook IIL
acquiring the firft form of that body> might by degrees quite
Ibfe their gravity. And therefore the weights would depend
upon the forms of bodies^ and with thpfe forms might be
changed : contrary to what was proved in the preceding
corollary.
Con. 3. All fpaces are not equally full ; for if all fpaoes
were equally full, then the fpecific gravity of the fluid
which fills the region of the air, on account of the extreme
denfity of the matter, would fall nothing (hort of the fpecific
gravity of quickiilver, or gold, or any other the moft denfe
body ; and, therefore, neither gold, nor any other body, could
defceod in air ; for bodies do not defcend in fluids^ unlefs
tbey are fpecifically heavier than the fluids. And if the quan-
tity of matter in a given fpace can, by any rarefaaion, be
diminiflied, what fliould hinder a diminution to infinity i
CoR. 4'. If all the folid particles of all bodies are of the
fame denfity, nor can be rarefied without pores, a void, fpace,'
or vacuum mufl; be granted. By bodies of the fame denfity, I
mean thofe whofe vires inertia are in the proportion of their
bulks.
Cor. 5. The power of gravity is of a different nature from
the power of magnetifm ; for tlie magnetic attra&icMi is not
as the matter attracted. Some bodies are attracted more by
the magnet ; others lefs ; moft bodies not at alK The power
of magnetifm in one and the fame body may be increafed
and diminiflied ; and is fometimes far ftronger, for the quan-
tity of matter, than the power of gravity ; and in receding
from the magnet decreafes not in the duplicate but almoft
in the triplicate proportion of the diftance, as nearly as I
could judge from fome rude obfervations.
PROPOSITION VII. THEOREM VII.
That there is a power of gravity tending to all bodies, propor*
Honal to the federal quantities of matter which they contain.
That all the planets mutually gravitate one towards another,
we have proved before ; as well as that the force of gravity
towards every one of them, confidered apart, is reciprocally
as the fquare of the difi;ance of places from the centre of the
fdanet. And thence (by prop. Gg, book 1, and its corollaries)
Book III. OF NATURAL 4?HILOSOPHY. 177
it follows^ that the gravity tending towards all the planets is
proportional to the matter which they contain.
Moreover, fii^ce all the parts of any planet A gravitate to-
wards any other planet B ; and the gravity of every part is to
the gravity of the whole as the matter of the part to the mart-
ter of the whole ; and (by law 3) to every a6)ion correfponds
an equal re-a6tion ; therefore the planet B will, on the other
hand, gravitate towards all the parts of the planet A ; and its
gravity towards any one part will be to the gravity towards the
whole as the matter of the part to the matter of the whole,
Q-E.D.
CoR. !• Therefore the force of gravity towards any whole
planet arifes frcxn, and is compounded of, the forces of gravity
towards all its parts. Magnetic and ele6);ric attra<^ons ai^^d
us examples of this; for all attra<^on towards t\ie whole arifes
from the attra6Uohs towards the feveral part3. The thing
may be eafily underftood in gravity, if we coniider a greater
planet, as formed of a number of lefTer planets, meeting to-
gether in one globe ; for hence it would appear that the force
of the whole muft. strife from the forces of the coinponeat
parts. If it is pbj^6l^,.t(iat, according to this law, all bodices
with us muft mutually .g-ravitate one towards another, whereas
no fuch gravitation any .where appears, 1 anfwer, that .fince
the gravitation towards thefe bodies is to the gravitation to-
wards the whole earth as th^fe bodies are to the whole earthy
the gravitation towards them muft be far lefs than to fall un-
der the obfervation of ppr fenfes.
CoR. 2. The force of gravity towards the feveral equal, pai:-
ticles of apy bpdy is reciprocally as the fquare of the dif-
ti^nce of .places.fropi the particles ; as appears from cor. S,
prop. 74, book 1. i
PROPOSITION VIII. THEOREM VIII.
Intwofpheres mutmUy gravitating each towards the other, if
the matter in places on all fides round about and equidiftant
Jromtht centres is fimilar, the weight of either fphere to-
wards the other will be reciprocally as the fquare of the dif
tance between their centres.
Yoh. U. N
178 MATHEMATICAL PRINCIPLES Book III.
After I had found that the force of gravity towards a whole
planet did arife from and was coooipounded of the forces of
gravity towards all its parts, and towards every one part was
in the reciprocal proportion of the fquarcs of the diftances
from the part, I was yet in doubt whether that reciprocal
duplicate proportion did accurately hold, or but nearly fo,
in the total force compounded of fo many partial ones ; for
it might be that the proportion which accurately enough
took place in greater diftances fhould be wide of the truth
•near the furface of the planet, where the diftances of the
particles are unequal, and their fituation diflSmilar. But by
the help of prop. 75 and 76, book 1, and their corollaries,
I was at laft fatisfied of the truth of the propoiition, as it now
lies before us.
Cob. 1. Hence we may find and compare together the
weights of bodies towards different planets ; for the weights
of bodies revolving in circles about planets are (by cor. 2,
prop. 4, book 1) as the diameters of the circles dired^iy,
and the fquares of their periodic times reciprocally ; and
their weights at the furfaces of the planets, or at any other
diftances from their centres, are (by, this prop.) greater or lefs
in the reciprocal duplicate proportion of the diftances. Thus
from the periodic times of Venus, revolving about die fon^
"in 224*. IGJ**. of the utmoft circumjovial fatellite revolving
about Jupiter, in 16*. l6^^. ; of the Huygenian fatellite about
Saturn in 15'*. 22f^. ; and of the moon about the earth in 27**.
7*^. 43' ; compared with the laean diftance of Venus from the
fun, and with the greateft heliocentric elongations of the out-
moft circumjovial fatellite from Jupiter's centre, 8' l6'' ; of
the Huygenian fatellite from the centre of Saturn, 3' 4'' ; and
of the moon from the earth, 10' 33": by computation I found
that the weight of equal bodies, iit equal diftances from the
centres of the fun, of Jupiter, of Saturn, and of the earth,
towards the fun, Jupiter, Saturn, and the earth, were one
to another, as 1, n?rf> tAt> ^^^ a^Sii refpeAively. Then
becaufe as the diftances are increafed or diminifbed, the
weights are diminifbed or increafed in a duplicate ratio, the
weights of equal bodies towards the fun, Jupiter, Saturn, iind
Hatexyil. Toll.
fla^f7»
ynfe^mum^ ScM/Wi0 fyM ^i
Book IIL OF NATURAL PHILOSOPHY* 179
the eertb^atthediftances 10000>d97^791> and 109 from their
centres, ,diai is at their very fuperficies, will be as 10000, 943,
699, and 435 refpedtively. How much the weights of bodies
are at the fuperficies of the moon, will be (hewn hereafter.
Cob. 2. .Hence likewife we difcover the quantity of matter
iii the feveral planets ; for their quantities of matter are as
the forces of gravity at equal diftances from their centres;
that is, in the fun, Jupiter, Saturi), and the earth, as 1, 77^^
r^TTf and i-t^Vtt refpedtively. If the parallax of the fun be
taken ^eater or lefe than lo" SO'", the quantity of matter
in the earth muft be augmented or diminifhed in the tripli<«
cate of that proportion.
Cor. 3. Hence alfo we find the denfities of the plaYiets ;
fox (by prop. 72, book 1) the weights of equal and ilmilar
bodies towards fimilar fpheres are, at the furfaces of thofe
fpheres, as the diameters of the fpheres ; and therefore the
denfities of diffimilar fpheres are as thofe weights applied to
the diameters of the fpheres. But the true diameters of the
fun, Jupiter, Saturn, and the earth, were one to another as
10000, 997^ 791> and 109; and the weights towards the fame
as 10000, 943, 529^ and 435 refpedlively ; and therefore their
denfities are as 100, 94|, 67, and 400. The denfity of the
earth, which comes out by this computation, does not depend
upon the paraUax of the fun, but is determined by the paral-
lax of the moon, and therefore is here truly defined. The
fun, therefore, is a little denfer than Jupiter, and Jupiter
than Saturn, and the earth four times denfer than the fun;
for the fun, by its great heat, is kept in a fort of a rarefied
ilate. The moon is denfer than the earth, as fhall appear
afterwards.
Cor. 4. The fmaller the planets are, they are, ceteris
paribus, of fo much the greater denfity ; for fo the powers
of gravity on their feveral furfaces come nearer to equality.
They are likewife, cateris paribus, of the greater denfity,
as they are nearer to the fun. So Jupiter is more denfe than
Saturn, and the earth tban Jupiter ; for the planets were to
be placed at different diflances from the fun, that, according
to their degrees of denfity, they might enjoy a greater or lefs
N a
ISd MATlH'BMATICAL PBIKCIPLES JBook 1II«
proportion of ihe ftin's heat. Our water, if H were removed
ajs fkr as the orb of Satarn, would be converted into ice, and
tti dre orb of Mecory would quidcly fly away in vapour; for
the light of the fun, to which its heat is proportional, is
feven times denfer in the orb of Mercury than with iis:
imd by the theimometer I have found that a fevenfold beat
of our fammer fun will make water boil. Nor are we to
dofubt tifafat the matter of Mercury is adapted to its beat, and
is therefore more denfe than the matter of our earth ; fince,
hi a denfer matter, the operations of Nature require a fironger
heat.
PROPOSITION IX. THEOREM IX.
I%al the force of gravity, confidertd downwards from the
furface cf the planeUy decreafes nearly in the proportion qf^
the diftancesfrom their centres.
If the matter of the planet were of an uniform denfity, this
propofitioh would be accurately true (by prop. 73, book 1)..
The error, therefore, can be no greater than what may
arife from the inequality of the denfity.
PROPOSITION X. THEOREM X.
That the motions of the planets in the heavens fnayfubfifi arm-
exceedingly long time.
In the fcbolium of prop. 40, book 2, I have ihewed tha^
a globe of water frozen into ice, and moving freely in our air,
in the time that it would defcribe the length of its femi-diame«
ter, would lofe by the refiftance of the air tsVt part of 'i\m
motion ; and the feme proportion holds nearly in all globes,
now great foever, and moved with whatever velocity. But
that our globe of earth is of greater denfity than it would be
if the whole confifted of water only, I thus make out. If the
whole confifted of water only, whatever was of lefs denfity
than ^ater, becaufe of its lefs fpecific gravity, would emerge
and float above. And upon this account, if a globe of ter-
refl^rial matter, covered on all fides with water, was left denfe
tfian water, it would emerge fomewhere ; and, the fubfiding
water falling back, would be gathered to the oppofite fide.
And fuch is the condition of our earth, which in a great
meafure is covered with feas. The earth, if it was tiot for
Book III. OF NATURAL PHILOSOPHY. 181
its greater denfity^ would emerge from the fea$^ and^ accord-
ing to its degree of levity^ would be raifed more or lefi aboye
their furface^ the water of the feas flowing, backwards to the
oppofite fide. By the fame argument^ the fpots of the fuQj
which float upon tbe lucid matter thereof^ are lighter than,
that matter; and^ however the planets have been formed
while they were yet in fluid mafles, all the heavier matter fub-
fided to the centre. Since, therefore, the common matter of
our earth on the furface thereof is about twice as heavy as
water, and a little lower^ in mines, is found about three, or
four, or even five times more heavy, it is probable that the
quantity of the whole matter of the earth may be five or fix
times greater than if it confifl:ed all of water; efpecially fince
I have before (hewed that the earth is about four times more
denfe than Jupiter. If, therefore, Jupiter is a little more denfe
than water, in the fpace of thirty days, in which that planet
defcribes the length of 459 of its femi«diameters, it would, in a
medium of the fame denfity with our air, lofe almoft a tenth
part of its motion. But fince the refiftance of mediums dci-
creafes in proportion to their weight or denfity, fo that water,
which^fi 13t times lighter than quickfilver, refifl:s lefs in that
proportion ;* and air, which is 860 times lighter than water,
refifts lefs in the fame proportion ; therefore in the heavens,
where the weight of the medium in which the planets move
is immenfely diminiflied, the refiftance will almoft vanifli.
It is fliewn in the fcholium of prop. 22, book 2, that at th^
height of 200 miles above the earth the air is more rare than
it is at the fuperficies of the earth in the ratio of 30 to
0,0000000000003998, or as 75000000000000 to 1 nearly.
And hence the planet Jupiter, revolving in a medium of the
fame denfity with that fuperior air, would not lofe by the refift-
ance of the medium the 1000000th part of its motion in 1000000
years. In the fpaces near the earth the refiftance is pro-
duced only by the air, exhalations, and vapours. When thefe
are carefully exhaufted by the air-pump from under the re-
ceiver, heavy bodies fall within the receiver with perfe<ft free-
dom, and without the leaft fenfible refiftance : golditfelf, and
the lighteft down, let fall together, will defcend with equal
N 3
IBt MATHEMATICAL PBINCIPLES Book IIL
velocity ; and though they fall through a fpace of fbur^ fix>
and eight feetj they will come to the bottom at the Tame
time; as appears from experiments. And therefore the
celeftial regions being perfeAly void . of air and exhalations^
the planets and comets meetings no fenfible refiflance in
thofe fpaces^ will continue their motions through them for an
immenfe tradl of time.
HYPOTHESIS I.
That the centre ofthefyftem of the world is immovable.
This is acknowledged by all, while fome contend that Hm.
earth, otTiers that the fun, is fixed in that centre. Let us fee
what may from hence follow.
PROPOSITION XI. THEOREM XI.
I%at the common centre of gravity of the earth, the fun, anS
all the planets, is immovable
For (by cor. 4 of the laws) that centre either is at reft, or
moves uniformly forward in a right line ; but if that centre
moved, the centre of the world would move alfo^ againft the
hypothefis.
PROPOSITION XII. THEOREM XII.
That the funis agitated by a perpetual motion, but never recedes
far from the common centre of gravity of all the planets.
For fince (by cor. 2, prop 8) the quantity of matter in the
^ fun is to the quantity of matter in Jupiter as 1067 to 1 ; and
the diftance of Jupiter from the fun is to the femi-diameter
of the fun in a proportion but a fmall matter greater, the
common centre of gravity of Jupiter and the fun will fall
upon a point a little without the furface of the fun. By
the fame argument, fince the quantity of matter in the fun is
to the quantity of matter in Saturn as 3021 to 1, and the
diftance of Saturn fi'om the fun is to the femi-diameter of the
fun in a proportion but a fmall matter lefs, the common
centre of gravity of Saturn and the fun will fall upon a point
a little within the furface of the fun. And, purfuing the
principles of this computation, we ihould find that though
the earth and all the planets were placed on one fide of the
fun, the difixmce of the common centre of gravity of all from
the centre of the fun would fcarcely amount to one diameter
Book III. . OF NATURAL PHILOI^OPHY* IBS
of the fun. In other cafes^ the diftances of thofe centres are
alvrays lefe ; and, therefore, fince that centre of gravity is in
perpetual reft, the fun, according to the various pofitions of
the planets, muft perpetually be moved every way, but will
never recede far from that centre.
Cob. Hence the common centre of gravity of the earth,
the fun, and all the planets, is to be efteemed the centre of the
world ; for fince the earthy the fun, and all the planets, mu-
tually gravitate one towards another, and are therefore, ac-
cording to their powers of gravity, in perpetual agitation, as
tbe laws of motion require, it is plain that their moveable cen-
tres cannot be taken for the immovable centre of the world.
If that body were to be placed in the centre, towards which
other bodies gravitate mpft (according to common opinion),
that privilege ought to be allowed to the fun ; but fince the
fun itfelf is moved, a fixed point is to be chofen from which
the centre of the fun recedes leaft, and from which it would
recede yet lefs if the body of the fun were denfer and greater,
and therefore lefs apt to be moved.
PROPOSITION XIII. THEOREM XIII.
ITie planets move in ellipfcs which have their common focus in
the centre of the fun; and, by radii drawn to that centre,
they defcribe areas proportional to the times of defcription.
We have difcourfed above of thefe motions from the phae-
nomena. Now that we know the principles on which they
depend, from thofe principles we deduce the motions of the
heavens a priori. Becaufe the weights of the planets towards
the fun are reciprocally as the fquares of their diftances from
the fun's centre, if the fun was at reft, and the other planets
did not mutually a<^ one upon another, their orbits would be
ellipfcs, having the fun in their common focus ; and they
would defcribe areas proportional to the times of defer iption,
oy prop. 1 and 11, and cor. 1, prop. 13, book 1. But the
mutual a6lions of the planets one upon another are fo very
fmall, that they may be negledled ; and, by prop. 66, book 1,
they lefs difturb the motions of the planets around the fun in
motion than if thofe motions were performed about the fun
at reft.
N4
184 MATlIEMATl^iVL iPlEtlllCIPLBd J^OdA lit*
' It 18 troe> that the aAion of Jupiter upon Saturn is not td
be neglefted ; for the force of gravity towards Jupiter is to the
force of gravity towards the fun as 1 to 106? ; and therefore
in the conjun6lion of Jupiter and Saturn, becaufe the dif-
tanceof Saturn from Jupiter is to the diftance of Saturn from the
fun abnoft as 4 to 9, the gravity of Saturn towards Jupiter
win be to the gravity of Saturn towards the fan as 81 to l6
X 1067 ; or, as 1 to about 211. And hence arifes a pertur-
bation of the orb of Saturn in every conjunAion of this
planet with Jupiter, fo fenfible, that aftronomers are puzzled
with it. As the planet is differently fituated in thefe con-
jun6lions, its eccentricity is fometimes augmented, fometimes
diminifhed ; its aphelion is fometimes carried foi*wards, fome-
times backwards, and its mean motion is by turns accelerat-
ed and retarded ; yet the whole error in its motion about
the fun, though arifing from fo great a force, may be ahnoft
avoided (except in the mean motion) by placing the lower
focus of its orbit in the common centre of gravity of Jupiter
and the fun (according to prop. 67, book 1), and therefore
that error, when it is greateft, fcarcely exceeds two minutes;
and the greateft error iti the mean motion fcarcely exceeds
two minutes yearly. But in the corijunftion of Jupiter and
Saturn, the accelerative forces of gravity of the fun towards
Saturn, of Jupiter towards Saturn, and of Jupiter towards the
r in ^ r. ■« 16X81X3021 ^^_
fun, are almoft as l6, 81, and > or 156609;
25
and therefore the difference of the forces of gravity of the
fun towards Saturn, and of Jupiter towards Saturn, is to the
force of gravity of Jupiter towards the fun as 65 to 156609,
or as 1 to 2409. But the greateft power of Saturn to difturb
the motion of Jupiter is proportional .to this difference ; and
therefore the perturbation of the orbit of Jupiter is much lefs
than that of Saturn's. The perturbations of the other orbits
are yet far lefs, except that the orbit of the earth is fenfibly
difturbed by the moon. The common centre df gravity of
the earth and moon moves in an ellipfis about the fun in the
focus thereof, and, by a radius drawn to the fun, defcribes
area* proportional to the times of defcription. But the earth
\
HI. ^ 6P NATURAL PHILOSOPHY. 185
in the mean time by a menftrual motion is revolved about tbia
common centre.
PROPOSITION XIV. THEOREM XIV.
The aphelions and nodes of the orbits of the planets are
fixed.
The aphelions are immovable, by prop. 1 1, book 1 ; and fo
are the planes of the orbits^ by prop. 1 of the fame book. And
if the planes are fixed, the nodes mnft be fo too. It is trae,
that fome inequalities ma(y arife from the mutual a6Uons of the'
planetd and comets in their revolutions ; but thefe will be fo
fmall, that they may be here pafled by.
CoR. 1. The fijted fiars are immovable, feeing they keep
the fame pofition to the aphelions and nodes of the planets.
Cor. 2. And fince thefe ftars are liable to no fenfible pa-
rallax from the annual motion of the earth, they can have no
force, becaufe of their immenfe diftance, to produce any fen-
fible effeft in our fyftem. Not to mention that the fixed fl;ani,
every where prcwnifcuoufly difperfed in the heavens, by their
contrary attradlions deftroy their mutual adions, by prop. 70,
book 1.
SCHOLIUM.
Since the planets near the fun (viz. Mercury, Venus, the
Earth, and Mars) are fo fmall that they can a6l but with little
force upon each other, therefore their aphelions and nodes
mufi; be fixed, excepting in fo far as they are difturbed by the
a6lions of Jupiter and Saturn, and other higher bodies. And
hence we may find, by the theory of gravity, that their aphe-
lions move a little in confequentiay in refpeft of the fixed fi:ars,
and that in the fefquiplicate proportion of their feveral difl;-
ances from the fun. So that if the aphelion of Mars, in the
l^ace of an hundred years, is carried 33' 20" in confequentia,
in re^a of the fixed ftars, the aphelions of the Earth, of
Venus, and of Mercury, will in an hundred years be carried
forwards 17' 40", 10' 53", and 4' l6", refpeftively. But
thefe motions are fo inconfiderable, that we have negle6ted
them in this propofition.
J86 MATHEMATICAL PRINCIPLES JBooA: IIL
PROPOSITION XV. PROBLEM I.
To find the principal diameters of the orbits of the planets^
They are to be taken in the fub-fefquiplicale proportion of
the periodic times, by prop. 15^ book 1, and then to be feve-
rally iaugmented in the proportion of the fum of the mafles of
matter in the fun and each planet to the firft of two mean
proportionids 4>etwixt that fum and the quantity of matter in
the fun^ by prop. 60> book 1 •
PROPOSITION XVI, PROBLEM II.
To find the eccentricities and aphelions of the planets.
This problem is refolved by prop. 18, book 1.
PROPOSITION XVIL THEOREM XV.
I%at the diurnal motions of the planets are uniform, and that
the lihration of the moon arifesfrom its diurnal motion.
The propoiition is proved from the firft law of motion, and
cor. 22, prop. 66, book 1. Jupiter^ with refpe<Sl to the fixed
ftars, revolves in 9^. 56' ; Mars in 24^. 39' ; Venus in about
£3*^; the Earth in 23^. 56' ; the Sun in 25| days, and the moon
in 27 days, 7 hours, 43'. Thefe things appear by the phasno-
mena. The fpots in the fun's body return to the fame fitu-
atipn on* the fun's diik, with refpe6l to the earth, in 27i days ;
and therefore with refped); to the fixed ftars the fiin revolves in
about 25| days. But becaufe the lunar day, arifing from its
uniform revolution about its axis, is menftrual, that is, equal
to the time of its periodic revolution in its orb, therefore the
fame face of the moon will be always nearly turned to the up-
per focus of its orb ; but, as the fituation of that focus requires,
will deviate a little to one fide and to the other from the
earth in the lower focus ; and this is the libration in longi-
tude ; for the libration in latitude arifes from the moon's la-
titude, and the inclination of its axis to the plane of the
ecliptic This theory of the libration of the moon, Mr. N.
Mcrcator, in his Aftronomy, publiflied at the beginning of the
year I676, explained more fully out of the letters I fent him.
Tlie utmoft fatellite of Saturn feems to revolve about its axis
with a motion like this of the moon, refpe6Ung Saturn con-
tinually with the fame face ; for in its revolution round Sa«
turn^ as often as it comes to the eaftem part of its orbit^ it is
Book IIL OF NATURAL PHILOSOPHY. 187
fcarcely vifible, and generally quite difappears ; which is like
to be occafioned by feme fpot3 in that part of its body^ which
is then turned toward the earthy as M. Caffini has obferved.
So alfo the utmoft fatellite of Jupiter feems to revolve about
its axis with a like motioo, becaufe in that part of its body
which is turned from Jupiter it has a fpot^ which always ap«
pears as if it were in Jupiter's own body, whenever the fatellite
paiTes between Jupiter and our eye.
PROPOSITION XVIII. THEOREM XVI.
That the axes of the planets are lefs than the diameters drawn
' perpendicular to the axes.
The equal gravitation of the parts on all fides would give a
fpherical figure to the planets, if it was not for their diurnal
revolution in a circle. By that circular motion it comes to
pafs that the parts receding from the axis endeavour toafcend
about the equator; and therefore if the matter is in a fluid
ftate, by its afcent towards the equator it will enlarge the dia-
meters there, and by its defcent towards the poles it will
ihorten the axis. So the diameter of Jupiter (by the concur-
ring obfervations of aftronomers) is found fliorter betwixt pole
and pole than from eafi; to wefi:. And, by the fame argument,
if our earth was not higher about the equator than at the
poles, the feas would fubfide about the poles, and, rifing to* '
wards the equator, would lay all things there under water.
PROPOSITION XIX. PROBLEM III.
To find the proportion of the axis of a planet to the diameters
perpendicular thereto.
Our countryman, Mr. Norwood, meafuring a diftance of
905751 feet of London meafure between London and York, ip
1635, and obferving the difference of latitudes to be Q^ 0,8',
determined the meafure of one degree to be 367 1 96 feet of
London meafure, that is 57300 Paris toifes, M. Picart, mea-
furing an arc of one degree, and 22' 55" of the meridian
between Amiens and Malvoifine, found an arc of one degree
to be 57060 Paris toifes. M. CaJJini, the father, meafured the
diftance upon the meridian from the town of Collioure in
Rouffillon to the Obfervatory of Paris ; and his fon added
the diftance from the Obfervatory to the Citadel of Dunkirk.
188 MATHEMATICAL PKIKCIPLEft Book III.
The whole difiance was 486l56f toifes^ and the difference of
the latitudes of Coiliourc and Dunkirk was 8 degrees^ and
31' liy". Hence an arc of one degree appears to be 5706 1
ParU ioifes. And from tbefe meafures we conclude that the
circumference of the earth is 123249600/ and its femi-dia-
meter ig6 15800 Paris feet^ upon the fuppofition that the
earth is of a fpherical figure.
In the latitude of Parts a heavy body falling in a fecond of
time defcribes 15 Parts feet, 1 inch, 1 line, as above, that
h, 9173 lines -f . The weight of the body is diminifbed by the
weight of the ambient air. Let us fuppofe the weight loft
thereby to be Tyhnr part of the whole weight ; then that
heavy body falling in vacuo willdefcribe a height of 2174 lines
in one fecund of time.
A body in every fidereal day of 23^. 56' 4" uniformly re-
volving in a circle at the diftance of 19615800 feet from the
centre, in one fecond of time defcribes an arc of 1433, 46
feet ; the verfed fine of which is 0,05236561 feet, or 7^4064
lines. And therefore the force with which bodies defcend in
the latitude of Paris is to the centrifugal force of bodies in the
equator arifing from the diurnal motion of the earth as 2 174
to 7,54064.
The centrifugal force of bodies in the equator is to the
centrifiigal force with which bodies recede direftly from the
earth in the latitude of Paris 48° 50' 10" in the" duplicate
proportion of the radius to the cofine of the latitude, that is,
as 7,54064 to 3,297- Add this force to the force with which
bodies defcend by their weight in the latitude of Pans, and a
1)ody, in the latitude of Paris, falling by its whole undimi-
niihed force of gravity, in the time of one fecond, will defcribe
2177,267 lines, or 15 Paris feet, 1 inch, and 5,267 lines.
And the total force of gravity in that latitude will be to the
centrifugal force of bodies in the equator of the earth as
2177,267 to 7,54064, or as 289 to 1.
Wherefore if APBQ (PL 10, Fig. I) reprefent the figure of
the earth, now no longer fpherical, but generated by the ro-
tation of an ellipfis about its leiTer axis ; and ACQqca a canal
fiill of water, reaching from the pole Qq to the centre Cc^
B^k in. OF NATURAL PHiLOSdPHY. id)
and tiience rrfing to the equator Aa ; the weight of the water
in the leg of the canal ACca will be to the weight of water in
the other leg QCcq as 289 to 288^ becaufe the centrifugal
force aiifibg from the circular motion fuftains and takes off
one of the £89 parts of the weight (in the one leg), and the
weight of £86 in the other fuftains the reft. But by compu-
tation (from cor. 2, prop, gi^book 1) I find, that, if the mat-
ter of the earth was all uniform, and without any motion, and
its axis PQ were to the diameter AB as 100 to 101, the ^orce
of gravity in the place Q towards the earth would be to
the force of gravity in the fame place Q towards a fpJbere
defcribed about the centre C with the radius PC, or QC« 93
126 to 125, And, by the fame argument, the force of gra*-
vity in the plabe A towards the fpheroid generated by the
rotation of the ellipfis APBQ about the axis AB is to the
foree of gravity in the fame place A, towards the fphere de-
fcribed about the centre C with the radius AC, as 125 to 126.
But the force of gravity in the place A towards the «arth is
a mean proportional betwixt the forces of gravity towards
that fpheroid and this fphere ; becaufe the fphere, by having
its diameter PQ diminiflied in the proportion of 101 to 100^
is transformed into the figure of the earth ; and this figure, by
having a third diameter perpendicular to the two diameters
AB and PQ diminifhed in the fame proportion, is converted
into the faid fpheroid ; and the force of .gravity in A, in either
cafe, is diminifhed nearly in the fame proportion. Therefore
the force of gravity in A towards the fphere defcribed about
the centre C with the radius AC, is to the force of gravity
in A towards the earth as 126 to 125|. And the force of
gravity in the place Q towards the fphere defcribed about
the centre C with the radius QC, is to the force of gravity in
the place A towards the fphere defcribed about the centre C,
with the radius AC, in the proportion of the diameters (by
prop. 72, book 1), that is, as 100 to 101. If, therefore, we
compound thofe three proportions 126 to 125, 126 to 1252^
and 100 to 101, into one, the force of gravity in Jhe place
Q towards the earth will be to the force of gravity in the
igO MATHEMATICAL PRINCIPLES Book IIL
place A towards the earth as 126 x 126 X 100 to 125 X
1254 X 101; or as 501 to '500.
Now fince (by cor. 3, prop. 91, book 1) the force of gravity
in either leg of the canal ACca, or QCcq, is as the diftance
of the places from the centre of the earth, if thofe legs are
conceived to be divided by tranfverfe, parallel, and equidiftant
furfaces, into parts proportional to the wholes, the weights of
any nnmber of parts in the one leg ACca will be to the
weights of the fame number of parts in the other leg
as their magnitudes and the accelerative forces of their
gravity conjunAly, that is, as 101 to 100,* and 500 to
501, or as 505 to 501 . And therefore if the centrifugal force
of every part in the leg ACca, arifing from the diurnal motion,
was to the weight of the fame part as 4 to 505, fo that from
the weight of every part, conceived to be divided into 505
parts, the centrifngal force might take off four of thofe parts,
the weights would remain equal in each leg, and therefore
the fluid would reft in an equilibrium. But the centrifugal
force of every part is to the weight of the fame part as 1 to
289 ; that is, the centrifugal force, which (hould be ^vv pstrts <^
the weight, is only -^Jr part thereof. And, therefore, I fay, by
the rule of proportion, that if the centrifugal force -yj^ make
the height of the water in the leg ACca to exceed the
height of the water in the leg QCcq by one -^ part of its
whole height, the centrifugal force -yfy will make the excefs
of the height in the leg ACca only -yfy part of the height
of the water in the other leg QCcq ; and therefore the
diameter of the earth at the equator, is to its diameter from
pole to pole as 230 to 229- And fince the mean femi-diameter
of the earth, according to Picarfs menfuration, is 19615800
Paris feet, or 3923,16 miles (reckoning 5000 feet to a mile),
the earth will be higher atthe equator than at the poles by
85472 feet, or 17yV miles. And its height at the equator
will be about 19658600 feet, and at the poles 19573000 feet.
If, the denfity and periodic time of the diurnal revolution
remaining the fame, the planet was greater or lefs than the
earth, the proportion of the centrifugal force to that of gra-
vity, and therefore alfo of the diameter betwixt the poles to
the diameter at the equator, would likewife remain the fame.
But if the diurnal motion was accelerated or retarded in any
Booklll. OV NATUBAL PHILOSOPHT. 191
proportion^ the centrifugal force would be augmented or
diminiihed nearly in the fame duplicate proportion; and
therefore the difference of the diameters will be increafqd or
diminifhed in the fame duplicate ratio very nearly. And if
the denfity of the planet was augmented or diminiflied in any
proportion^ the force of gravity tending towards it would alfo
be augmented or diminilhed in the fame proportion; and the
difference of the diameters contrariwife would be diminifhed
in proportion as the force of gravity is augmented) and aug-
mented in proportion as the force of gravity is diminifhed.
Wherefore^ fince the earthy in refpe6); of the fixed flars^ re*
volves in 23^. 5&, but Jupiter in 9*. 56*, atid the fquares
of their periodic times are as £9 to 5, and their denfities
as 400 to 94i, the difference of the diameters of Ju-
^400 229
piter will be to its leffer diameter as 5 X 941 x 1 to 1, or
as 1 to 9}., nearly. Therefore the diameter of Jupiter
from eafl to weflis to its diameter from pole to pole nearly as
10j< to 9|. Therefore fince its greateit diameter is 37'^ its
lefTer diameter lying between the poles will be 33" 25'". Add
thereto about 3" for the irregular refradlion of light, and the
apparent diameters of this planet will become 40" and 36"
25"'; which are to each other as Hi to lOf, very, nearly.
Thefe things are fo upon the fuppofition that the body of
Jupiter is uniformly denfe. But pow if its body be denfer
towards the plane of the equator than towards the poled, its
diameters may be to each other as 12 to 11, or 13 to 12, or
perhaps as 14 to 13.
And Cajjini obferved^ in year I691, that the diameter of
Jupiter reaching from eafl to wefl is greater by about a
fifteenth part than the other diameter. Mr. Pound with
his 123 feet telefcope, and an excellent micrometer, meafured
the diameters of Jupiter in the year 1719, and found them as
follow.
The Times. | Greateft diam. I Leffisr diam. | The diam. to each other.
;;; — ; ^ ij«__- L^ ' • -
As 12 to 11
1S| to 12i
129 to ]1|
14| to 13|
Da)r. Hours.
January 28 6
March 6 7
March 9 7
April g 9
— i '
Parts.
13,40
13,12
13,12
12,32
Faru.
12,28
12,20
12,08
192 MATHEMATICAL PRIMCIPI.ES Book IQL
So that the theory agrees with the phflenomena ; for the
l^anets are more heated by the fun's rajs towards their equa-
torsy and therefore are a little more condenfed by that heat
Ihan towards their poles.
Moreover^ that there is a diminution of gravity occaConed
iy the diomal rotation of the earthy and therefore the earth
rifes "higher there than it does at the poles (fuppofing that its
matter is uniformly denfe), will appear by the experiments of
pendulums related under the following proportion.
PROPOSITION XX. PROBLEM IV.
To find and compare together tJie weights of bodies in ihedif--
ferent regions of our earth,
Becaufe the weights of the unequal legs of the canal of*
water ACQqca are equal ; and the weights of the parts poor^
portional to the whole legs^ and alike fituated in them^ are one
to another as the weights of the wholes^ and theoefoie aqoijL
betwixt themfelves ; the weights of equal parts^ and alike
fituated in the legs^ will be reciprocally as the legs^4hat is^
feciprocally as £30 to 8^9. And the cafe is the fame in aU.
Iiomogeneous eqnal bodies alike fituated in the legs of the
canal. Their weights are reciprocally as the legs^. that is^
reciprocally as the diflances of the bodies from the centre of
the earth. Therefore if the bodies are fituated in the upper-'
mofi; parts of the canals^ or on the furface of the -earthy their*
weights will be one to another reciprocally as their difiance^
from the centre. And^ by the fame argument^ the weights
in all other places round the whole furface of the earth are
reciprocally as the diftances of the places from the <;entre ;
and, therefore, in the hypothefis of the earth's being a fphe-
roid, are given in proportion.
Whence ariies this theorem, that the increafe of .weight
in paffing from the equator to the poles is nearly aS' the verlied
fine of double the latitude; or, which comes to the iame thing,
Bs the fquare of the right fine of the latitude ; and tlie arcs
of the degrees of latitude in the meridian increafe nearly in
.the fame proportion. And, therefore, fince the latitude of
Paris is 48° 50', that of places under the eqiiator 00° 00', aud
that of places under the poles gO° ; and the verfed fines of
double Uiofe arcs are 11334^00000 and 20000^ the radius be-
Book llh OF NATI7RAL PHILOSOPHY. 193
ing lOCXX); and the force of gravity at the pole is to the force
of gravity at the equator as 230 to 229; and the excefs of
the force of gravity at the pole to the force of gravity at the
equator as 1 to 229 ; the excefs of the force of gravity ia
the latitude of Paris will be to the force of gravity at
the equator as 1 X UUt to 229> or as 566? to 2290000.
And therefore the whole forces of gravity in thofe places
will be one to the other as 2295667 to 2290000. Where-
fore iince the lengths of pendulums vibrating in equal
times are as the forces of gravity^ and^ in the latitude of Paris^
thelengthof a pendulum vibrating feconds is 3 Paris feet^ and
6i lines, or rather, becaufe of the weight of the air, 84. lines^
the length of a pendulum vibrating In the fame time under
the equator will be Ihorter by 1,087 lines. And by a Uke
calculus the following table is made.
Latitude of
Length of the
Meafure of one degxee
the place.
pendulum.
in the meridian.
Dcgr.
Feet. Lines.
Toifei.
3 . 7,468
56637
*»
3 • 7,482
56642
10
3 . 7,526
56659
15
3 . 7,596
56687
20
3 . 7,692
56724
25
3 . 7,812
56769
30
3 . 7,948
56823
35
3 . 8,099
56882
40-
3 . 8,261
56945
1
3 . 8,294
56958
2
3 . 8,327
56971
3
3 . 8,361
56984
4
3 . 8,394
56997
45
3 . 8,428
57010
6
3 . 8,461
57022
7
3 . 8,494
57035
■ 8
3 . 8,528
57048
9
3 . 8,561
57061
50
3 . 8,594
57074
55
3 . 8,756
57137
60
3 , 8,907
57196
65
3 • 9/>44
57250
70
3 . 9,162
57295
75
3 . 9,258
57332 .
80
3 • 9,329
57360
85
3 . 9,372
57377.
90
3 . 9,387
57382
U.
)g4 MATHEMATICAL ^KINClPlES Sook III.
Bj thi^ table, therefore, it appears ihdt the b^qmdrty of
degrees is (b fmaH, thai the figure of the earth, iii geograpbl*
fcal mattan, may be confidered as fpherical ; efpeeitilly if the
earth be a little denfer towards the plane of the equator than
towards the poles.
Now feveral aftronomers, fent into remote coan tries to make
aftnmomicalobferTations, have found that pendulum cloeks do
aecordmgly move flQWer near the equator than in our climates.
And, firft of all, in the year 167^^ M. Richer took notice of
it in the ifland of Cayenne ; for when, in the month of uf tf-
guft, he was obferring the tranfits of the fixed tksn over the
meridian, he found his clock to go flower than it ooghl in re-
tpe€t of the nieati motion of the fun at the rate of S^28" m day.
Therefore^ fitting up a fimple pendulum to vibrate fai fecod^,
which were meafured by an excellent clock, he obfieirved the
length of that fimple pendulum ; and this he did over and
over every week for ten months together. And upon his
letom to France, eompaHng the length of that pendnlnoi
with the length of die pendukmt at Paris (which was 3 Paris
feet and 8f iixK&% he found it fhorter by 1 i Ifne.
Afterwards, our fnend I^r. Halley, about the year lG77i ar*
riving at the ifland of St. Heleny found his pendulum clock to
go flower there than at London, without marking the difier^
ence. But he fhortened the rod of his clock by more than
the |. of an inch, or li Ine ; and to effect this, becaufe the
length of the fcreW at the lower end of the rod was not
f nfficient, he interpofed a wooden ring betwixt the nut and
the ball.
. Then, in the year 1689, M. Farin and M. 4es Hayes found
the length of a .fimple peadulum vibrating in feconds at the
Royal Obfervatoiy of Par& to be S feet and 8} lines. And fay
the fame method in the ifland ofGoree, they found the length
of an ifochronal pendulum to be 3 feet and 6|. linds, differ-
ing from the former by two lines. And in tte fame year, go*
ing to the iflands of Guadaioupe and Mariinko, they found
that the length of an ifochronal pendolutn in tfaofe iflimds was
3 feet aad 61 lines*
Book m. OF KATtTEAL FHXlOSOPHT. IQS
After tbis, M, Couplet, the fon, in the month o(Julyl6Q7, at
the Rojral Obfervatorj of Paris, fo fitted his pendulum clock
to the mean motion of the fun^ that for a confiderable time
together the clock agreed with the motion of the fun. In
tiovember following^ upon his arrival at Lijbon, he found his
ekx^k to go flower than before at the rate of 9l 13" in 24
hoars. AxA next March coming to Paraiba, he found his
clock to go flower there than at Paris, wfd at the rate 4' I2f
in £4 hours ; and he affirms^ that the pendulum vibrating
in feconds was ftuMtter at Li/bon by 2| line^ and at Paraiba
fay Sx lines, than at Paris. He had done better to have
lemoned thofe difierences 1^ and ^; for thefe differences
ccnrrefpond to the differences of the times i IS" and 4' \^\
But this g^itieman's obfervations are fo grofs^ that we cao-
not confide in them. ■<
In the following years^ 1699 And 170a, M. des Hayes,
making another voyage to America, determined that in the
Slands of Cayenne and Granada the length of the pendolum
tibrating in feconds was a fmall matter lefs than 3 feet and
6| lines ; that in the ifland of St. Chriftophers it was 3 feet
and 6i lines ; and in the ifland of St. Domingo S feet and 7
lines.
And in the year 1704, P. Teuiili, at Puerto Bello in
America, found that the length of the pendulum vibrating in
feconds was 3 Park feet, and only 5^ lines, that is, almoft
S lines &orter than at Paris ; but the obfervation was faulty*
For afterwards, going to the ifland of Martimco, he found the
length of the iibchronal pendulum there 3 Paris feet and
^Hnes.
Now the latitude ot Paraiba b & 38' fouth ; that of Puerto
Bello 9^ SS' north ; and the latitudes of the iflands Cayenne^
l^Qfee, Gaudaloupe, Martimco, Granada, St. Chrijiophcrs^
ind St. Domingo, ore refpeaively 4'' 55', 14^ 4ff, J 4** 00', 14^
14'/I«f»ee', 17** 19', a«d 19^ 48', north. And the exoeffes of
;he leng& of the pendulum at Paris above the lengths of the
ibchronal pendulums obferved in thofe latitudes are a little
preater than by the table of the lengths of the pendulum
before computed. And therefore the earth is a little higher
O 2
igd MATHEMATICAL PRINCIPLES Book III.
under the equator than by the preceding calcuhis^ and a little
denfer at the centre than in mines near the furface^ unlefs^
perhaps^ the heats of the torrid zone have a little esrtended
the length of the pendulums.
: For M . Picart has obferved, that a rod of iron, which in
frofty weather in the winter feafon was one foot long, when
heated by fire, was lengthened into one foot and | line. Af-
terwards M. de la Hire found that a rod of iron, which in the
like winter feafon was 6 feet long, when expofed to the heat
of the fummer fun, was extended into 6 feet and f line. In
the former cafe the heat was greater than in the latter ; but
in the latter it was greater than the heat of the external parts
df a human body ; for metals expofed to the fummer fun.
acquire a very confiderable degree of heat. But the rod of
a pendulum clock is never expofed to the heat of the fum-
mer fan, nor ever acquires a heat equal to that of the external
parts of a human body ; and, therefore, though the 3 feet
rod of a pendulum clock will indeed be a little longer in the
fummeir than in the winter feafon, yet the difference will
fcarcely amount to j line. Therefore the total difference of
^e lengths of ifochronal pendulums in different climates can-
not be afcribed to the difference of heat ; nor, indeed, to the
jniftakes of the French aflronomers. For although there is
not a perfe6l agreement betwixt their obfervations, yet the
errors are fo fmall that they may be negle6led ; and in this
they all agree, that ifochronal pendulums are ihorter under
the equator than at the Royal Obfervatory of Paris, by a
flifference not lefs than l| line, nor greater than 2f lines. By
the obfervations of M. Bicker in the ifland of Cayenne, thQ
difference was 1^ Une. That difference being corre6led by
^hofe of M. des Hayes, becomes l| line or 1| line. By th^
Jefs accurate obfervations of others, the fame was made about
two lines. And this difagreement might arife partly from the
errors of the obfervations, partly from the diffimilitude of the
internal parts of the earth, and the height of mountains;
partly from the different heats of the air.
. I take an iron rod of 3 feet long to be fhorter by a fixth
part of one line in winter time with us here in England , th^m
in the fummer. Becaufe of. the great heats under the
I'lau xxmj^ii.
liaffewS.
1
1Bl>w<*r<v( tftr^'it'ftf-i ' V. f
• ^
JioOk HT. OF NATURAL PHILOSOPHY: 197
equajtor^ filbdiuSl this quantitj front the difFerence of one line
apd a quarter obferved by M. Richer, and there will remain
one line ^y which agrees very well with l-j4^ line colle<%ed^
by the theory a little before. M. Richer repeated his obfer-
rations^ made in the ifland of Cayenne, every week for ten
months together^ and compared the lengths of the pendulum
which he had there noted in the iron rods \^ith the lengths
thereof which he obferved in France. This diligeitce and
care feems to have been wanting to the other obfervers. If
this gentleman's obfervations are to be depended on^ the earth
is higher under the equator than at the polesj, and that by aa
excefs of about 17 miles ; as appeared above by the theoiy.
PROPOSITION XXI. THEOREM XVII.
That the equinoSial points go backwards, and that the axis of
the earthy by n nutation in every annual revolution, twice
vibrates towards the ecliptic, atid as often returns to its for^
merpofition.
The propofition appears from cor. 9,0, prop. 66, book 1 ;
but that motion of nutation mud be very fmall, and, indeed^
fcarcely perceptible.
PROPOSITION XXII. THEOREM XVIII.
That all the motions of the moon, and all the inequalities of
thofe motions, follow from the principles which we have laid
dorxm.
That the greater planets, while they are carried about the
fun, may in the mean time carry other leffer planets, revolv-
ing about them ; and that thofe leffer planets muit move in
ellipfes which have their foci in the centres of the greater,
appears from prop. Q5, book 1. But then their motions will
be feveral ways drflurbed by the a6lion of the fun, and they
will fuffer fuch inequalities as are obferved in our moon.
Thus our moon (by cor. 2, 3, 4, and 5, prop. 66, book 1) moves
fafler, and, by a radius drawn to the earth, defcribes an area
greater for the time, and has its orbit lefs curved, and ther^
fore approaches nearer to the earth in the fyzygies than in the
quadratures, excepting in fo far as thefe effe^b are hindered
by the motion of eccentricity ; for (by cor. 9, prop. 66, book
J) the eccentricity is greateft when the apc^eon of the moon*
3
]g8 MATHBMATICAL FHINCIFLEU Book IlL
If in tbe fyzygiesj and leaft when the fame is in the qnadr a^
tares ; and upon this account the perigeon moon is fwifter^
and nearer io os^ but the apogeon moon Hower, apd fiurtbei*
irom u%, in tbe fyzygies than in the quadratures. More*
orrer^ the apogee goes forwards^ and thenodes backwards ; and
Ibis is done not with a regular but an unequal motion. For
(t^^ cor. 7 and 8, prop, 66^ book 1) the apogee goes more (Wiffly
fimrards in its fyzygies^ more flowly backwards in its quadra*
lures ; and^ by the excefs of its progrefs aboYC its r^^refs, ad-
Tances yearly in eonfequentia. But^ contrariwife^ the nodes
(by cojf. 11^ prop. 66, book 1) are quiefcent in their fyrjrgies,
and go fafteft back in their quadratures. Farther^ the greafceft
latitude of the moon (by cor. 10^ prop. 66, book I) is greater
in the quadratures of the moon than in its fyzygies. And (by
cor. 6, prop. 661, book 1) the mean motion of the moon |s flower
in the perihelion of the earth than in its aphelicm. And
thefe are the principal inequalities (of the moon) taken notice
of by aftronbmers.
But there are yet other inequalities not obferved by fonner
aftronomers^ by which tbe motions of the moon are fo di£*
turbed, that to this day we have not been able to bring them
under any certain rule. For the velocities or horary motions
of the apogee and nodes of the moon, and their equations^ as
well as the difference betwixt the greateft eccentridty in
the fyzygies, and the leaft eccentricity in the quadratures,
and that inequality which we call the variation, are (by
cor. 14, prop. 66, book 1) in the courfe of the year aug-
mented and diminiihed in the triplicate proportion of the
fun's apparent diameter. And befides (by cor. 1 and 2, lem.
10, and cor. 16, prop. 66, book 1) the variation is augmented
and diminifhed nearly in the duplicate proportion of the time
between the quadratures. But in aftronomical calculations
this inequality is commonly thrown into and confounded with
the equation of the moon's centre,
PROPOSITION XXIII. PROBLEM V.
To derive the unequal motiom of the fatelliies of Jupiter and
Saturn from the motions of our moon.
From the motions of our moon we deduce the correi^nd-
ing motions of the moons or fatelUtes of Jupiter in this man-
Hook III. OF NATOftAt VmhOSOfUY. " 1^9
net, by edi. l6, prpp. 66^ book 1. The mean motion of &o
nodes of the outmoil fatellite of Jupiter ib to the meaa motioa
of the nodes of our moon in a proportion compounded of the
duplicate proportion of the periodic time of the earth abo^
the fun to tbe periodic time of Jupiter abopt the fun, and ti^
fimple proportion of the periodic time of the fatcjii^e aboitf
Jupiter to the periodic time of our nv>on about the earth ; and*
therefore, thofe nodes, in tlie fpace of a hundred years, 9X%
carried 9^ 9Al backwards, or in antecedentia. The mean mo«
tionit of the nodes of the inner fatellites are to the mean mo^
tion of the nodes of the outmofl as their periodic times to the
periodic tune of the former, by the fame corollary, aod ar^
thence ^ven. And the motion of the apfis of every fatdliit^
in cmfoquentia is to the motion of its nodes in antecedentia
as the motion of the apogee of our moon to the motion of its
nodes (by the fame corollary), and is thence given. But the
motions of the apfides thus found muft be diminiflied in the
proportion of 5 to 9> or of about 1 to 2,>on account of a caufe
which I cfmnot here defcend to explain. The greateft equa*
tions of the nodes, and of the apiis of every fatellite, are to
the greateft equations of the nodes, and apogee of our mopo
refpe6Uvely> as the motions of the nodes and apfides of the
fatellites, in the time of one revolution of the former equ^y
tions, to the motions of the nodes and apogee of our moon,
in the time of one revolution of the latter equations. The va-
riation of a fatellite feen from Jupiter is to the variation of our
9K>on ia. the fame proportion as the whole motions of their
nodes refpe&ively during the times in which the fatellite and
our moon (after parting from) are revolved (again) to the fun,
by the fame corollary ; and therefore in the outmofl fatellite
the variation does not exceed 5" 12'".
PROPOSITION XXIV. THEOREM XIX.
J'hai thefiux and reflux of the fea arifefrom the a&iom-of the
fun and moon.
By cor. 19 and £0, prop. ^, book 1, it appears that the
waters of the fea ought twice to rife and twice to fall every
day, as well lunar as folar ; and that the greateft height of
the waters in the open and deep feas ought to foUoir the ap«
04
^06 mrAl^kMATICAL PRINCIPL-ilS Book IIL
pxxKe of the liiminaries to the meridian of the pladr by a left
interval than 6 hours ; as happens in all that eaftem tradl of
the Atlantic and Mthiopic feas between France and the Cape
of Good Hope; and on the coafts of Chili BXiA Perti in the
South Sea ; in all which (hores the flood falls out about the
feiscond^ thirds or fourth hour^ unlefs where the motion pro-
pagated from the deep ocean is by the fliallownefs of the
channels^ through which it paiTes to fome particular places^
retarded to the fifth, iixth, or feventh hour, and even later.
The hours I reckon from the appulfe of each luminary to the
meridian of the place, as well under as above the horizon ; and
by the hours of the lunar day I underftand the 24th parts of
that time which the moon, by its apparent diurnal motion,
employs to come about again to the meridian of the place
which it left the day before. The force of the fun or moon
in raifing the fea is greateft in the appulfe of the luminary
to the meridian of the place ; but the force impreffed upon
the fea at that time continues a little while after the impref-
fion, and is afterwards increafed by a new though lefs force
ilill a6);ing upon it. This makes the fea rife higher and higher,
i;ill this new force becoming too weak to raife it any more, the
fi^a rifes to its greateft height. And this will come to pafs>
perhaps, in one or two hours, but more frequently near the
ihores in about three hours, or even more, where the fea is
Ihallow.
The two luminaries excite two motions, which will not ap-
pear diftindlly, but between them will arife one mixed motion
compounded out of both. In the conjun6lion or oppofition
of the luminaries their forces will be conjoined, and bring on
the greateft flood and ebb. In the quadratures the fun will
raife the waters which the moon deprefles, and deprefs the
waters which the moon raifes, and from the difference of their
forces the fmalleft of all tide^ will follow. And becaufe (as
experience tells us) the force of the moon is greater than that
of the fun, the greateft height of the waters will happen about
the third lunar hour. Out of the fyzygies and quadratures,
the greateft tide, which by the fingle force of the moon ought
to fall out at the third lunar hour, and by the fingle force of
the fun at the third folar hour, by the compounded forces of
SdoRltL OP KATURAL PHILOSOPHY. QOt
both miift%iU out in an intermediate time, that approached
nearer to the third hour of the moon than to that of the fuq.
Andy therefore^ while the moon is paffing from the fyzygiet
to the quadratures^ during which time the 3d hour of the fun
precedes the 3d hour of the moon^ the greatefl height of tike
waters will alfo precede the 3d hour of the moon^ and that^
by the greateft interval^ a little after the o6);ants of the moon ;.
andy by like intervals^ the greateft tide will follow the 3d lu-
nar hour^ while the moon is paflfing froifn the quadratures to
the fyzygics. Thus it happens in the open fea ; for in the
mouths of rivers the greater tides come later to their height.
ButtheeiFedlsofthe luminaries depend upon their diftances
from the earth ; for when they are lefs diftant^ their eiFe6U
are greater^ and when more diftant, their ej9e(3^ are lefs^ and
that in the triplicate proportion of their apparent diameter.
Therefore it is that the fun, in the winter time, being then
in its perigee, has a greater eSe&., and makes the tides in th^
fyzygies fomething greater, and thofe in the quadratures fome-
thing lefs than in the fummer feafon ; and every month the
moon, while in the perigee, raifes gi'eater tides than at the
difiance of 15 days before or after, when it is in its apogee*
Whence it come to pafs that two faigheft tides do not follow
one the other in two immediately fucceeding fyzygies.
The effe6t of either luminary doth likewife depend upon its
declination or diftance from the equator ; for if the luminary
was placed at the pole, it would conftantly attradl all the partf
of the waters without any intenfion or remiiiion of its a(^on,
and could caufe no reciprocation of motion. And, therefore^
as the luminaries decline from the equator towards either pole>
they will, by degrees, lofe their force, and on this account
will excite lefler tides in the folftitial than in the equino<9:ial
fyzygies. But in the folftitial quadratures they will raife greater
tides than in the quadratures about the equinoxes ; becaufe
the force of the moon, then fituated in the equator, moft ex-
ceeds the force of the fun. Therefore the greateft tides fall
out 5n thofe fyzygies, and the leaft in thofe quadratures, which
happen about the time of both equinoxes : and the greateft
tide in the fyzygies b always fucceeded by the leaft tide ia
•n MATBBMATICAL niMClPLSt B$0k UL
the qnadntnret, at we find by expemooe. Bat, becaiife the
fiin 18 lefs diftaotfrom th^ earth in winter than in foouner, i&
comet U> pafs that the gieateft and kaft tides moie ftaqoentlj
appear before than after the vernal equinox^ and nuve be^
qiently aftar than before the aotomnal.
Moreover, the effeds of the lominariet dqpend upon the
latitudes of places. Let ApEP (PL 10, Fig. S) repiefent the
• earth covered with deep waters ; C its centre ; P, p its poles ;
A£ the equator ; F any fdace without the equator ; Ff the
parallel of the place ; Dd the correfpoodent parallel on the
other fide of the equator; L the place of the moon thiee boors
belbre ; H the place of the earth direAIy under it; h the op-
pofiie place ; K^ k the places at QO degrees diftanoe ; CHj
Ch, the greateft heights of the fea from the centre of the
earth ; and CK, ck, ils leaft heights : and if with the axes
. Hh, Kk, an ellipfis is defcribed, and by the revokdion of that
dlipfis about its longer axis Hh a fpheroid HPKfapk is
formed, this fpheroid will nearly reprefent the figure of the
lea ; and CF, Cf, CD, Cd^ will leprefent the h^hto of the
fea in the places Ff, Dd. But farther ; in the laid revoln*
tton of the ellipfis any point N defcribes the circle NM cnt<
ting the parallels Ff, Dd, in any places RT, and the equator
AE in S ; CN will reprefent the height of the fea in all thofe
places R, S, T, fituated in this circle. Wherefore, in the di-
urnal revolution of any place F, the greateft flood will be in
F, at the third hour after the appulfe of the moon to the m^i-
dian above the hcNrizon ; and afterwards the greateft ebb in Q^
at the third hour after the fetting of the moon ; and then the
greateft flood in f, at the third hour after the appulfe of the
moon to the meridian under the horizon ; and, lafily, the
greateft ebb in Q, at the third hour after the rifing of the moon ;
and the latter flood in f will be leis than the preceding flood
m F. For the whole fea is. divided into two hemifpherical
floods, one in the hemifphere KHk on the north fide, the
other in the oppofite hemifphere Khk, which we may there-
fore call the northern and the foathem floods. Thefe floods,
being always oppofite the one to the other, come by turns to
the meridians of all places, after an interval of 12 lunar hours.
Book HL op NATURAL PHILOSOPHY. 203
And feeing the norlbern countries partake more of the nor* .
them floods and the fouthern coantries more of the foathem
floods thence arife tides, alternately greater and lefs in all
places withodt the equator^ in which the luminaries rife and
fet* Bat the greateft tide will happen when the moon de*
clines towards the vertex of the place, about the third hour after
the i^polfe of the moon to the meridian above the horizon ;
and when the moon changes its declination to the other fide of
the iquator, that which was the greater tide will be changed
inlD « lefler. And the greateft difference of the floods will fall
oat mboat the times of the folflices ; efpecially if the afcend*
ing node of the moon is about the firfi of Aries. So it is found
by experience that the morning tides in winter exceed thofe
of the evening, and the evening tides in fummer exceed thofe
of the morning ; at Plymouth by the height of one foot, but
at Bri/lol by the height of 15 inches, accoi-ding to the obfer*
vattons of Colepreft and Sturmy.
Bat the motions which we have been defcribing fuffer fome
alteratioa from that force of reciprocation, which the waters,
bring once moved, retain a little while by their vis infita^
Whence it comes to pafk that the tides may continue for fome
time^ though the a6i;ions of the luminaries fhould ceafe. This
power of retaining the imprelTed, motion lefTens the difference
of the alternate tides, and makes thofe tides which imme-
diatdy fncceed after the fyzygies greater, and thofe which
follow next afler the quadratures lefs. And hence it is that
the alternate tides at Plymouth and Brijiol do not differ much
more one from the other than by the height of a foot or 15
inches, and that the greatefl tides of all at thofe ports are not
the firft but the third after the fyzygies. And, befides, all the
motions are retarded in their paffage through fhallow chan-
nelsj fo that the greatefl tides of all, in fome flraits and
months of rivers, are the fourth or even the fifth after the fy-
zygies.
Farther^ it may happen that the tide may be propagated
from the ocean through different channels towards the fame
port^ and may pafs quicker through fome channels than
tbroigh others; in which cafe the fame tide> divided into two
fi04 MATHEMATICAL PRINCIPLES BookHH
or more fucceedbg one another, may compound new mo-
tions of different kinds. Let us fuppofe two equal tides flow-
ing towards the fame port from different places, the one pre-^
ceding the other by 6 hours ; and fuppofe the firft tide to
happen at the third hour of the appulfe of the moon to the
meridian of the port. If the moon at the time of the appnlfe
to the meridian was in the equator, every 6 hours alternately
there would arife equal floods, which, meeting with as many
equal ebbs, would fo balance one the other, that for thai day
the water would ftagnate and remain quiet. If the moolatfaen
declined from the equator, the tides in the ocean would be
alternately greater and lefs, as was faid ; and from thence two
greater and two leffer tides would be alternately propagated
towards that port. But the two greater floods would make
the greateft height of the waters to fall out in the middle time
betwixt both ; and the greater and lefler floods would make
the waters to rife to a mean height in the middle time between
them, and in the middle time between the two leffer floods
the waters would rife to their leafl; height. Thus in the fpace.
of 24 hours the waters would come, not twice, as commonly,
but once only to their greatefl, and once only to their leail
height ; and their greateft height, if the moon declined to-
wards the elevated pole, would happen at the 6th or SOth hour
after the appulfe of the moon to the meridian ; and when the
moon changed its declination, this flood would be changed into
an ebb. An example of all which Dr. Halley has given us,
from the obfervations of feamen in the port of Bat/ham, in the
kingdom of Tunquin, in the latitude of 20*^ dOf north. In that
port, on the day which follows after the paffage of the mooiv
over the equatoi', the waters fl:agnate : when the moon declines
to the north, they begin to flow and ebb, not twice, as in
6'ther ports, but once only every day ; and the flood happens
at the fetting, and the greatefl; ebb at the rifing of the moon.
This tide increafes with the declination of the moon till the
7th or 8th day ; then for the 7 or 8 days following it decreafes
at the fame rate as it had increafed before, and ceafes when
the moon changes its declination, crofling over the equator
to the fouth. After which the flood is immediately chaiiged
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Book UK OF NATURAL PHILOSOPHY. ^05
into an ebb; and thenceforth the ebb happens at the.fettuig
and the flood at the riiing of the moon ; till the moon^ agaia
p^og the equator^ changes its declination. There are two
inlets to this port and the neighbouring channels^ one from
the feas of ChmUj between the continent and the ifland of
Jjuamia ; the other from the Indian fea, between the conti-
nent and the ifland of Borneo. But whether there be really
two tides propagated through the faid channels^ one from the
Indian fea in the fpace of 12 hours^ and one from the fea of
CUna in the fpace of 6 hours^ which therefore happeoing at;
the 3d and gth lunar hours^ by being compounded together^
produce thofe motions ; or whether there be any other cir-
cumftances in the fiate of thofe feas^ I leave to be determine
ed by obfervations on tlie neighbouring fhores.
Thus I have explained the caufes of the motions of the
moon and of the fea. Now it is fit to fubjoin fomething con-*
ceming the quantity of thofe motions.
PROPOSITION XXV. PROBLEM VI.
Tojind the forces with which the fun dijiurbs the motions ofth^
moon. (PI. 10, Fig. 3.)
Let S leprefent the fun, T the earth, P the moon, CADB
the moon's orbit. In SP take SK equal to ST ; and let SI4
be toSK in the duplicate proportion of SK to SP: draw LM
parallel to FT ; and if ST or SK is fuppofed to reprefent the
accelerated force of gravity of the earth towards the fun, SL
will leprefent the accelerative force of gravity of the moon
towards the fun. But that force is compounded of the parti
SM and LM, of which the force LM, and that part of SM
which is reprefented by TM, difturb the motion of the moon,
as we have fliewn in prop. 66, book 1, and its corollaries*
Forafinuch as the earth and moon are revolved about their
common centre of gravity^ the motion of the earth about
that centre will be alfo difturbed by the like forces ; but we
inay confider the fums both of the forces and of the motiom
Itf in the moon, and reprefent the fum of the forces by th^
lines TM and ML^ which are analogous to them both. Th$
force ML (in its mean quantity) is to the centripetal forc^
by wbiclpi the moon may be retained in its orbit revolving
nt
the tarib wt icA, wt the d^Moe FT, in die
pioportjoo of the p e ri odic tone of the sMMift Jwfr die
to the periodic Ine of the CHth ahoot the Ab (by ear. 17f
piop. fiS, book 1); thai m, m thed^dksle piopo rti oe of tT^*
7^. 45" to 5651'. <?. gT; or OS 1000 to 178725; or oi 1 to
]78tt- Bet in the 41b prop, of this book «e fiMOid, tint, if
both csth aad aooo rae lerohned oboot thdr coouBOB oeft-
tie of gnrity, the Mean difiance of the ooe firom the oAor
vooU be oeariy a>{ mean fend-dianieiefsof theeaidi; «M
the finroe bj vhicfa the moon Bar be kqiC icfiil f iug in ita
orbiiaboot the earth in left at the diftaaoe FT of fll>{
diaanetcn of the earth, is to die fofce bj which ii auy be
lolrrd in thr fimr iiiar , at llii ilifl mif nf ftl ttmi diaii Ini^
as6t>fto60; and this fisroeis to thefaroeof grm^ widiaa
wiyneariyaa 1 tofio X 60. Theiefisfe the mean fbree ML
kto thefinroeof gnnrity onthefviaceofoarearthaa 1 X
60|to60xe0x60x l78H.or»l tofiMOgS/S; wfaoieo
hj the proportion of the Ii oca TM, ML, the fime TM is alfo
given ; and thefe are the fioroes with which the fan difaoba
the motions of the raooo. Q.E.L
PROPOSITION XXVI. PROBLEM VH.
To fimd the horary imcramaU ^ikt arta m?hich the aiaoii, Igr n
radims drmwm to the earthy defcnhet ta m dradmr orHtm
We have above (hewn thai the area which the moon de«
Icribes by a radios drawn to the earth is p iupor li onal to die
time of defcription, excepting in fo far as the moon's motion
is diftmbed by the adion of the fon ; and here we propofe to
inveftigate the inequality oF the moment, or horary inaenieut
of that area or motion fo difimrhtd. To render the cakttkHi
m<Ne eafyy we (hall foppofe the oibit of die moon to be cma-
lar, and neglcA all inequalities hot that only which is now
nnder confideration ; and, becaufeof the immenfe difiance of
the fan, we (hall^faEtber foppofe that the lines SP and ST
are parallel. By this means, the force LM (PI. 10^ fig. 4)
irill be always redaced to its mean qnantity TP^ as* well as
the force TM to its mean quantity 3PK. Thefe forces (by
cor. 2 of the laws of moti(m) compofe the force TL ; aiMtthis
force^ by letting &11 the perpendicular I£ upon the an^UvA
Book IIL OV KATtmAL PHILOSOPHY. d07
TP, it idblved into the forces TE, £L ; of which the force
TE> BtSting conftantly in the direction of the radius TP^ neither
accelerfttet nor retards the defcriptlon of the area TPC made
bjr that nuUas TP ; hot EL, a&ing on the radius TP in a per*
pendicnlar dire^ion, accelerates or retards the defcription of-
the arem in proportion as it accelerates or retards the moon.
That acceleration of the moon, in its paiTage from the qua*
dratore C to the conjnn&ion A, is in every moment of time
sPKxTK V
as ihe generating accelerative force EL, that is, as ■ ^p •'
Let the time be reprefented by the mean motion of the moon,
or (which, comes to the fame thing) by the angle CTP, or
even by the arc CP. At right angles upon CT ereft CG equal
to CT ; and, fuppofing the quadrantal arc AC to be divided
into aa infinite number of equal parts Pp, Sec. thefe part$
may leprefent the like if^nite number of the equal parts of
time. Let fall pk perpendicular on CT, and draw TG meet-
ing with KP, kp prodoced in F and f ; then will FK be equal
to TK> and Kk be to PK as Pp to Tp, that is, in a given pro«
portion; and therefore FK x Kk, or the area FKkf, will b«
sPK X TK
as ppp ■ , that is, as EL; and compounding, the whole
aiea GCKF will be as the fum of all the forces EL impreiTed
upon the moon in the whole time CP ; and therefore .alfo
88 the velocity generated by that fum, that is, as the accelera-
tion of the defcription of the area CTP, or as the incre*
ment of the moment thereof. The force by> which the^
moon may in its periodic time CADBof 97^. 7^. 43^ be
letaiiied revolving about the earth in reft at the diftance TP,
would caufe a body falling in the time CT to defcribe the
lei^fa -}Cr, and at the fame time to acquire a velocity equal
io diat with which the moon is moved in its orbit. This ap^
petim from cor. 9, prop. 4, book 1. But fince Kd, drawn per*
piendKctrlar on TP, is but a third part of EL, and equal to the
hdf of TP, or ML, in the oAants, the force EL in the o6hints,
llhere it is greateft, will exceed the force ML in the propor*
Ucto of 3 to 2 ; and therefore will be to that force by which
fliettoonia its periodic time may be ret&ined revcdving abottt
flp8 MATHEMATICAL i^RINCIPLES Book III.
Uie earth, at reft as 100 to f x 17872^, or 1 1915 ; and in the
time CT will generate a velocity equal to iHr ^ parts of the
velocity of the moon ; but in the time CPA will generate a
greater velocity in the proportion of CA to CT or TP. Let
the greateft force EL in the o6):ants be reprefented by the area
PK X Kk, or by the rediangle 4TP X Pp, which is equal
thereto ; and the velocity which that greateft force can ger*
jnerate in any time CP will be to the velocity which any other
lefier force EL can generate in the fame time as the re6);angle
TP X CP to the area KCGF; but the velocities generated
in the whole time CPA will be one to the other as the rect-
angle 4TP X C A to the triangle TCG, or as the quadrantal
arc CA to the radius TP; and therefore the latter ve-
locity generated in the whole time will be xirrr pstrt^
of the velocity of the moon. To this velocity of the
xnoon^ which is proportional to the mean moment of the
area (fuppoiing this mean moment to be reprefented by the
number 11915)^ we 4Eidd and fubtradl the half of the other
yelocity ; the fum 11915 + 50, or II965, will reprefent the
greateft moment of the area in the fyzygy A; and the differr
ence 11915-^50, or 11865, the leaft moment thereof in
the quadratures. Therefore the areas which in equal times
are. defcribed in the fyzygies and quadratures are one to
the other as 11 965 to II860. And if to the leaft moment
1 1865 we add a moment which (hall be to 100, the difference
of the two former moments, as the trapezium FKCG to the
triangle TCG, or, which comes to the fame thing, as the
fquare of the fine PK to the fquare of the radius TP (that is,
as Pd to TP), the fum will reprefent the moment of the area
when the moon is in any intermediate plac^ P.
But thefe things take place only in the hypothefis that
the fun and the earth are at reft, and that the fynodical
revolution of. the moon is finiflied in 27**. 7**. 43'. But fince
the. moon's fynodical period is really 29^. 12**. 44', the incre-
xnents of the moments muft be enlarged in the fame propor-
tion as the time is, that is, in the proportion of 1080853 to
JOOOOOO. Upon which account, the whole increment, whicl^
wag vi?i7 parts of the mean moment, will now become
ytSI^ parts therepf; and therefore the moment of the area
Book III. OF NATUBAL PHILOSOPHY. 209
in the quadrature of the moon will be to the moment
thereof in the fjzygy as 11023 — 50 to 11023 + 60; or as
10973 to ^1073 ; and to the moment thereof^ when the moon
is in any intermediate place P^ as 10973 to 10973 + Pd ;
that is, fuppofing TP = 1 00.
The area, therefore, which the moon, by a radius drawn
to the eardi, defcribes in the feveral little equal parts of time^
is nearly as the fum of the number 219,46, and the verfed
fine of the double diftance of the moon from the neareft
quadrature, confidered in a circle which hath unity for its
radius. Thus it is when the variation in the octants is in its
mean quantity. But if the variation there is greater or lefs,
that verfed fine muft be augmented or diminiflied in the fame
proportion.
PROPOSITION XXVII. PROBLEM VIIL
From the horary motion of the moon to find its diftance from
the earth.
The area which the moon, by a radius drawn to the earth,
defcribes in every moment of time, is as the horary motion of
the moon and the fquare of the diftance of the moon from the
earth conjundUy. And therefore the diftance ofi^^e moon
from the earth is in a proportion compounded of the fubdu-
plicate proportion of the area diredlly, and the fubduplicate
proportion of the horary motion inverfely. Q.E.I.
CoR. 1. Hence the apparent diameter of the moon is given;
for it is reciprocally as the diftance of the moon from the
earth. Let aftronomers try how accurately this rule agrees
with the phsendmena.
CoR. 2. Hence alfo the orbit of the moon may be more
exa6lly defined from the phsenomena than hitherto could be
done.
. PROPOSITION XXVIII. PROBLEM IX.
TTofind tfie diameters of the orbit, in z^hich, without eccen^
tricity, the moon would move.
The curvature of the orbit which a body defcribes, if at-
traded in lines perpendicular to the orbit, is as the force of
attra&ion diredly, and the f(|uare of the velocity inverfely*
I eftimate the curvatures of lines compared one with another
according to the evanefcent proportion of the fines or tan-
VoL. II. P
410 mathbmati<:;ai/ I^RiilciPLES JBwklTt..
gents of their angles ef eontadt to eqaal radii^ fuppofing thaCW
radii to be infinitely diminilhed. But the attraction of th<
moon towards the earth in the fjzygies is the esc^a of s
gravity towards the earth above the force of the fan 2^
(fee Fig. prop. 25), by which force the accelerativc gravi
of the moon towards the fun exceeds the aecelerative gravi
of the earth towards the fun, or is exce^ed by it. But
the quadratures that attra6lion is the fum of the gravity c:>-f
the moon towards the earth, and the fun*s force KT, by whic^tb
the moon is attradied towards the earth. And thefe attra<5lio
. . - ^ AT + CT ' 178725 . 200O
puttmg N for ^ > aie nearly as -jfjsT + gp ^
, 178725 1000 ^^ ^^ ^ ..w-..
and -y^+ atITN^ '''' "^ 178725N x CT* ^ SOOOA.TT'
X CT, and 178725N X AT* + IGOOCT* X AT. For if
the accelerative gravity of the moon towards the earth be r^s-
prefented by the number 178725, the mean force ML, whic^'^
in the quadratures is PT or TK> and draws the moon towax^^
the earthy will be 1000, and the mean force TM in the fy^JT'
gies will be 3000 ; from which, if we fubtraA the mean foro^
ML, there will remain 2000, the forc^ by which the mooU
in the fyzygies is drawn from the earth, and which we above
called 2PK. But the velocity of the moon in the fyzygi^*
A and B is to its velocity in the quadratures C and D as CTF
to AT, and the moment of the area, which the moon by ^
radius drawn to the earth defcribes in the fyzygies, to the mo-
ment of that area defcribed in the quadratures conjunftly »
that is, as 11073CTto 1097 3AT. Take this ratio twice in-
verfely, and the former ratio once diredly, and the curvature
of the orb of the moon in the fyzygies will be to the curvature
thereof in the quadratures as 120406729 X 178725A'P
X CT* X N— 120406729 X 2000AT* X CT to 12261 I52g
X 178725AT* x CT* x N + 122611329 X lOOOGT*
X AT, that is, as2151969AT x CT x N — 24081AP to
2191371AT X CT X N + l226lCT^
Becaufe the figure of the moon's orbit is unknown, let us,
in its Head, affume the ellipfis DBCA (PI. 10, Figl 5), in the
centre of which we fuppofe the earth, to be fituatiedj^ aad the
J%^i?XXV.T^/I.
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Book m. OF NAT1}RAL PHILOSOPHY. 211
greater axis de to lie between the quadratures as the leflet
AB between the fyzygtes. But fince the plane of this ellipfis
is revolved about the earth by an angular motion^ and th^
orbit, wbofe curvature we now examine, fhould be defcribed
in a plane void of fuch motion, we are to confider the figure
which the moon, while it is revolved in that ellipfis^ defcribes
in this plane, that is to fay, the figure Cpa, the feveral points
p of which are found by aiTuming any point P in the ellipfis,
which may reprefent the place of the moon, and drawing Tp
equal to TP in fuch manner that the angle PTp maybe equal
to the apparent motion of the fun from the time of the laft
quadrature in C ; or (which comes to the fame thing) that the
angle CTp may be to the angle CTP as the time of the fy-
Bodic revolution of- the moon to the time of the periodic revo-
htion thereof, or as 29'*. 12^. 44' to 27**. 7^. 43'. If, therefore,
in this proportion we take the angle CTa to the right angle
CTA, and make Ta of equal length with TA, we fhall have
a the lower and C the upper apfis of this orbit. But, by
computation, I find that the difference betwixt the curvature
of this orbit Cpa at the vertex a, and the curvature of a circle
defcribed about the centre T with the interval TA, is to the
difierence betwixt the curvature of the ellipfis at the vertex
A, and the curvature of the farhe circle, in the duplicate pro-
portion of the angle CTP to the angle CTp ; and that the cur-
tafnre of the ellipfis in A is to the curvature of that circle in
the duplicate proportion of TA to TC ; and the curvature of
that circle to the curvature of a circle defcribed about the
centre T with the interval TC as TC to TA ; but that the cur-
vature of this lajl arch is to the curvature of the ellipfis in C
in the duplicate proportion of TA to TC ; and that the dif-
ference betwixt the curvature of the ellipfis in the vertex C,
and the curvature of this lafl; circle, is to the difierence be-
hrixt the curvature of the figure Tpa, at the vertex C, and
the curvature of this fame laft circle, in the duplicate propor-
tion of the angle CTp to the angle CTP ; all which propor-
tions are eafily drawn from the fines of the angles of contact,
and of Jfit differences of thofe angles. But, by comparing
thofe pibportioDB together, wefind thq cmvature of the figure
P2
i
fit MATHEMATICAL PRIMCIPLSS BookJlh
Cpa at a to be to iU curvature at C as AT. X ^'f^^CT^AT
*o CP + ittnftWATx CT; where the number i^AftJW re-
prefents the difference of the fquares of the angles CTP and
CTp, applied to the fquare of the lefTer angle CTP ; or (which
is all one) the difference of the fquares of the times 27'. 7^. 43f,
and 29^. 12**. 44', applied to the fquare of the time 27^. 7^. 43'.
Since, therefore^ a reprefents the fyzygy of the moon, and
C itsjquadrature, the proportion now found mufl be the lame
with that proportion of the curvature of the moon's orb in
the fyzjgies to the curvature thereof in the quadratures,
which we found above. Therefore, in order to find the pro-
portion of CT to AT, let us multiply the extremes and the
means, and the terms which come out^ applied to AT x CT,
become 2062,79CT* — 2151969N x CT' + 368676N x
AT X CP + 36342 AT* X CT* — 3G2047N x AT* x CT
+ 219137 iN X AP + 4051,4AT* = 0. Now if for t|ie
half fum N of the terms AT and CT we put 1, and x for
their half difference, then CT will he = 1 + x, and AT =
1 — X. And fubftituUng thofe values in the equation^ after
refolving thereof^ we Ihall find x = 0,00219 9 stnd from
thence the femi' diameter CT =: 1,007 19> and the femi-dia-
meter AT =: 0,99281, which numbers are nearly as 70^^, and
69^. Therefore the moon's diftance from the earth in the
fyzygies is to its diftance in the quadratures (fetting aiide the
coniideration of eccentricity) as 69it to 70^; or, in round
numbers, as 69 to 70.
PROPOSITION XXIX. PROBLEM X.
To find the variation of the moon.
This inequality is owing partly to the elliptic figuce of
the moon's orbit, partly to the inequality of the moments of
the area which the moon by a radius drawn to the earth de-
fcribes. If tlie moon P revolved in the ellipfis DBCA about
the earth quiefcent in the centre of the ellipiis^ and by theradius
TP, drawn to the earth, defcribed the area CTP, proportional
• to the time of defcription; and the greateft femi-diameter
CT of the ellipiis was to the leaft TA as 70 to 69 ; the tan-
gent of the angle CtP would be to the tangent of the angk
of the mean motion, computed from the quadratore Cj as
Book III. OF NATURAL PHILOSOPHT. 213
the femi-diameter TA of the ellipfis to its femi-diameter TC,
or as 6d to 70. But the delcription of the area CTP, as the
moon advances from the quadrature to the fyzygy, ought to
be in fuch manner accelerated^ that the moment of the area
in the moon's fyzygy may be to the moment thereof in its
quadrature as 11073 to 10973; and that the excefs of the
moment in any intermediate place P above the moment in
the quadrature may be as the fquare of the fine of the angle
CTP; which we may effeil with accuracy enough, if we
diminiih the tangent of the angle CTP in the fubduplicate
proportion of the number 10973 to the number 11073, that
is, in proportion of the number 68,6877 to the number 69.
Upon which account the tangent of the angle CTP will now
be to the tangent of the mean motion as 68,6877 to 70 ; and
• the angle CTP in the oftants, where the mean motion is 45%
will be found 44"* 27' 28", which fubtraAed from 45^ the an-r
gle of the mean motion, leaves the greateft variation 32' 32".
Thus it would be, if the moon, in pafling from the quadrature
to the fyzygy, defcribed an angle CTA of 90 degrees only.
But becaufe of the motion of the earth, by which the fun is
apparently transfered in confequentia, the moop, before it over-
takes the fun, defcribes an angle CTa, greater than a right
angle, in the proportion of the time of the fynodic revolution
of the moon to the time of its periodic revolution, that is, in
the proportion of 29^ 12^44' to 27**. 7^ 43'. Whence it
Gomes to pafs that all the angles about the centre T are di-
lated in the fame proportion; and the greateft variation, which
otherwife would be but 32' 32'', now augmented in the faid
proportion, becomes 35' 10".
And this is its magnitude in the mean diftance of the fun
from the earth, n^gleding the differences which may a^ife
from the curvature of the orbis magnus, and the ftronger ac-
tion of the fun upon the moon when horned and new, than
when gibbous and frill. In other diftances of the fun from
the earth, the greateft variation is in a proportion compound-
ed of the duplicate proportion of the time of the fynodic re-
volution of the moon (the time of the year being given) di-
re&Ij, and the triplicate proportion of the diftance of the fun
PS
il^ MATHEMATICAL FKINCIPI^BS Book LXI*
from the earth inverfely. And^ therefore^ io the apogee of
the fun, the greateft variation is 33' H", aod in its perig^ee
87' 1 1", if the eccentricity of the fun is to the trapfverfe fero^-
diameter of the orbis magnum as l6ff- to. 1000.
Hitherto we have inveftigated the variation in anprhnot
eccentric^ in whicb> to wit^ the' moon in its o^lants is alwajift
in its mean diftance from the earth. If the moon> on ao*
count of its eccentricity, is more or lefs removed from ito
earth than if placed in this orb, the variation may be fome-'
thing greater, or fomething lefs, than according to this rule.
But I leave the excefs or defed to the determinatioq of afiro-
nomers from the phaenomena.
PROPOSITION XXX. PROBLEM XI.
Tojind the horary motion of the nodes of the moon in a circu'
lar orbit, (PL 11, Fig. 1.)
Let S reprefent the fun, T the earth, P the moon, NPn the
orbit of the moon, Npn the orthographic prc^edion of the
orbit upon the plane of the ecliptic ; N, n the nodes, nTNm
the line of the nodes produced indefinitely ; PI, PK perpendi-
culars upon the lines ST, Qq ; Pp a perpendicular upon the
plane of the ecliptic ; A, B the moon's fyzygies in the plane of
the ecliptic ; AZ a perpendicular let fall upon Nn, the line of
the nodes ; Q, q the quadratures of the moon in the plane of
the ecliptic, and pK a perpendicular on the line Qq lying be-
tween the quadratures. The force of the fun to difturb
the motion of the moon (by prop. 25) is twofold, one propor-
tional to the line LM, the other to the line MT, in the
fcheme of that propofition ; and the moon by the former
force is drawn towards the earth, by the latter towards the
fun, in a direcJlion parallel to the right line ST joining the
earth and the fun. The former force LM acls in the direc*
tion of the plane of the moon's orbit, and therefore makes no
change upon the fituation thereof, and is upon that account
to be neglected ; the latter force MT, by which the plane of
the moon's orbit is difturbed, is the fame with the force 3PK
or 3IT. And this force (by prop. 25) is to the force by which
the moon may, in its periodic time, be uniformly revolved in a
circle about the earth at reft, as 3lT to the radius of the circle
fliUe XJ^m. /mart/A .
#
Hook IIL PP NATUBAIi PHILOSOPHY. £15
multiplied by the number 178^725^ or as IT to the radius
thereof multiplied by 59,575. But in this calculus^ and all
that foliowty I conilder all the lines drawn from the moon to
the fuB as parallel to the line which joins the earth and the
fan ; beCBttfe what inclination there is almoft as much dimi-
nifhes ail <dfe6b in fome cafes as it augments them in others ;
and we are now enquuing after the mean motions of the
nodes, negle&ing fuch niceties as are of no moment, and
would only ferve to render the calculus more perplexed.
Now fupf^fe PM to reprefent an arc which the moon de-^
fcsribes in the leail moment of time, and ML a little line, the
balf of which the moon, by the impulfe of the faid force SlT,
would defciibe in the fame time ; and joining PI^ MP, let
them be produced to m and 1, where they cut the plane of
the ediptie, and upon Tm let fall the perpendicular PH.
T^im, finoe the right line ML is parallel to the plane of the
ecliptic, and therefore can never meet with the right line ml
which lies in that plane, and yet both thofe right lines lie in
oae common plane LMPml, they will be parallel, and upon
that account the. triangles LMP, ImP will be iimilal'. And
feeing MPm lies in the plane of the orbit, in which the moon
did move while in the place P, the point m will fall upon the
line Nil, which paifes through the nodes N, n, of that orbit.
And becaufe the force by which the half of the little line LM
is generated, if the whole had been together, and at once
lofprelTed rn the point P, would have generated that whole
line, and caufed the moon to move in the arc whofe chord is
LP ; that is to fay, would have transfered the moon from the
plane MPmT into the plane LPIT; therefore the angular
motion of the nodes generated by that force will be equal to
the angle mTl. But ml is to mP as ML to MP ; and fince
MP, becaufe of the time given, is aifo given, ml will bt- as
the reftangle ML X mP, that is, as the re6langle IT X mP.
ir ]
And if Tml is a right angle, the angle mTl will be as prr-.'
IT X Pm
and therefore as — yp 9 that is (becaufe Tm and inF, TP
P4
2X6 MATHEMATICAL PR1NCIPLB8 Book 111.
ITxPH
and PH are proportional), as — rpp ■ J andj, therefore^ be-
caufe TP is given, as IT x PH. But if the angle Tml or
STN is oblique, the angle mTl will be yet lefs, in proportion
of the fine of the angle STN to th^ radius, or AZ to AT.
And therefore the velocity of the nodes is as IT X PH x AZ,
or as the folid content of the fines of the three angles TPI,
PTN, and STN.
If thefe are right angles, as happens vtrhen the nodte are in
the quadratures, and the moon in the fyzygy, the little line ml
will be removed to an infinite diftance, and the angle mTI
will become equal to the angle mPL Bat in this cafe the
angle mPl is to the angle PTM, which the moon in the fame
time by its apparent motion defcribes about the earth; as 1 to
59,575. For the angle mPl is equal to the angle LPM, that
is, to the angle of the moon's deflexion from a re^linear path ;
which angle, if the gravity of the moon (boald have then
* ceafed, the faid force of the fun 3lT would by itfelf have ge-
nerated in that given time ; and the angle PTM is equal to
the angle of the moon's deflexion from a rectilinear path;
which angle, if the force of the fun SiT fliould have then
ceafed, the force alone by which the moon is retained in its
orbit would have generated in the fame time. And thefe
forces (as we have above fliewn) are the one to the other as I
to 59,575. Since, therefore, the mean horary motion of the
moon (in refpeft of the fixed ftars) is 32' 56" 27'" 12f ^ the
horary motion of the node in this cafe will be 33" itf" 33^\
12''. But in other cafes the horary motion will be to 33" 10'"
33K )2^ as the folid content of the fines of the three angles
TPI, PTN, and STN (or of the diftances of the moon from the
quadrature, of the moon from the node, and of the node from
the fun) to the cube of the radius. And as often as the fine
of any angle is changed from pofitive to negative, and from
negative to pofitive fo often muft the regreflive be changed
into a progreflive, and the progreffive into a regreflive motion.
Whence it comes to pafs that the nodes are progreflive ^s
often as the moon happens to be placed between either qua*
drature^ and (he node nearefl; to that quadrature. In Qther
Book IlL OF NATURAL PHILOSOPHY. £17
cafes they are regreffi ve^ and^ by the excefs of the regrefs above
this progrefs^ they are monthly transferee! in antecedentia,
' Cor. 1. Hence if from P and M, the extreme points of a
leaft arc PM (PI. 11, Fig. 2), on the line Qq joining the
quadratures we let fall the perpendiculars PK, M k, and pro*
duoe the fame till they cut the line of the nodes Nh in D
and d> the horary motion of the nodes will be as the area
MPDd, and the fquare of the time AZ conjundtly. For let
PK, PH,and AZ, be the three faid fines, viz. PK the fine of
the diftance of the moon from the quadrature, PH the fine
of the diflance of the moon from the node, and AZ the fine
dfthe diflance of the node from the fun; and the velocity
of the node will be as the folid content of PK X PH x AZ.
But PT is to PK as PM to Kk ; and, therefore, becaufe PT
and PM are given, Kk will be as PK. Likewife AT ia to
PD as AZ to PH, and therefore PH is as the re<ftangle PD
X AZ '; and, by compounding thofe proportions, PK x PH is
as the folid content Kk X PD x AZ, and PK x PH x AZ
as Kk X PD X AZ*; that is, as the area PDdM and AZ*
conjundly. Q.E.D.
Cor. 2. In any given pofition of the nodes their mean
horary motion is half their horary motion in the moon's
fyzygies; and therefore is to l6" 3o" l6*^ Sff". as the fquare
, of the fine of the diftance of the nodes from the fyzygies to
the fquare of the radius, or as AZ* to AT*. For if the moon,
by an uniform motion, defcribes the femi-circle QAq, the fum
of all the areas PDdM, during the time of the moon's
pafla^e firom Q to M, will make up the area QMdE, termi-
nating at the tangent QE of the circle; and by the time that
the moon has a/rived at the point n, that fum will make up
the whole area EQAn defcribed by the line PD: but when
the moon proceeds from n to q, the line PD will fall without
the circle, and will defcribe the area nqe, terminating at the
tangent qe of the circle; which area, becaufe the nodes were
before regreffive, but are now progreffive, muft be fubduc^-
ed from the former area, and, being itfelf equal to the
area QEN, will leave the femi-circle NQAn. While, there-
fore, the moon defcribes a femi-circle^ the fum of all the
areas PDdM will be the area of that femi-circle ; and while
318 MATHKICATICAL PEIlfCIPLCS Book Uh
the mooo defcribes a complete circle, the fam of thofe areas
will be the area of the whole circle. Bat the area PDdM^
when the mooo is io the fjzygkes, is the redaDgle of the arc
PM into the radium VT; and the fom of all the areas, ^veiy
ome equal to this area, io the time that the moon deicribes %
complete circle, is the redangle of the whole circamfereDoe
iaio the radius of the circle ; and this redangle, being ckmUe
the area of the circle, will be douUe the quantity ef M^e fbt^
mer fom. If, therefore, the nodes went on with that yelocilf
vBiformly continued which they acquire in the momi's lysjr*
giesy they would defcribe a fpace double of that which tbey
defcribe in fadi; and, therefore, the mean motion, by whicli,
if uniformly continued, they would defcribe the fame Ipace
with that which they do in fa6l defcribe by an unequal mo^
tion, is but one half of that motion which they are poflefled of
in the moon's fyzygies. Wherefore fince their greateft btxcarj
motion, if the nodes are in the quadratures, is 35" l(f S3^\
18^. their mean horary moticm in this cafe will be 16^
35"' iff". SfT. And feeing the horary motion of the
nodes is every where as AZ^ and the area PDdM coojuiidly^
and, tberefc»re, in the moon's fyzygies, the horary motion of
the nodes is as AZ* and the area PDdM conjun^y, that h§
(becaufe the area PDdM defcribed in the fyzygies is given)^
as AZ% therefore the mean motion alfo will be as AZ^; and^
therefore, when the nodes are without the quadratures, tbia
motion will be to 16' 36"' }&\ SfT. as AZ* to AT*, Q.E.D.
PROPOSITION XXXL PROBLEM XII.
To find the horary motion of the nodes of the moon in an elliptic
orbit. (PI. 12, Fig. 1.)
Let Qpmaq reprefent an ellipiis defcribed with the greater
axis Qq, and the kfler axis ab; QAqB a circle circumfcribed ;
T the earth in the common centre of both ; S the fun; p the
moon moving in this ellipfis ; and pm an arc which it deicribes
in the leaft moment of time; N and n the nodes joined by the
line Nn ; pK and mk perpendiculars upon the axis Qq, pro*
duced both ways till they meet the circle in P and M, and
the line of the nodes in D and d. And if the moon, by a
radim drawn to the earth, defcribes an area proportional to the
Tiate XI.T^/ JI.
/lrr^e2i S.
nt /
Book nh OF NATUAAL PHILOSOPHY. 219
time of defcfiption, tke horary motioa of tbe node in the
ellipiis will be as the area pDdm and AZ* conjnn<SIy.
For let PF touch the circle in P, and produced meet TN
in F; and pf touch the ellipfis in p^ and produced meet tbe
fame TN in f, and both tangents concur in the axis TQ at
¥• And let ML reprefent the fpace which the moon^ by
the impulfe of the above-mentioned force 3lT or SPK, would,
defcribe with a tranfverfe motion^ in the mean tim^ while
levolviDg in the circle it defcribes the arc PM ; and ml denote
tbe fpace which the moon revolving in the ellipfis woul4
defcribe in the fame time by the impulfe of the fame force
3lT or SPK ; and let LP and Ip be produc*ed till they naeet
tbe plane of the ecliptic in G and g^ and F6 and fg be joined,
of which FG produced may cut pf, pg, and TQ, in c, e, and
R refpedively ; and fg produced may cut TQ in r. Becaufe
the force 3lT or SPK in the circle is to tbe force 3lT or 3pK
iQ the dlipfis as PK to pK, or as AT to aT, the fpace
ML generated by the former force will be to the fpace ml
generated by the latter as PK to pK ; that is, becaufe of the
fimilar figures PYKp and FYRc, as FR to cR. Biit (becaufe
of the fimilar triangles PLM, PGF) ML is to FG as PL to
PG, that is (on account of tbe parallels Lk, PK^GR), as pi to
pe, that is (becaufe of the fiinilar triangles plm, cpe), as Im to
ce; and inverfely as LM is to Im, or as FR is to cR, fo is FG
to ce. And therefore if fg was to ce as fy to cY, that is, as
fr to cR (that is, as fr to FR and FR to cR conjunctly, that
is, as fT to FT, and FG to ce conjundly), becaufe the ratio
of FG to ce, expunged on both fides, leaves the ratios fg to
FG and fT to FT, fg would be to FG as fT to FT; and,
therefore, the angles which FG and fg would fubtend at
the earth T would be equal to each other. But thefe
angles (by what we have (hewn in the preceding propofition)
are the motions of the nodes, while the moon defcribes in tbe^
circle the arc PM, in the ellipfis the arc pm ; and therefore the
motions of the nodes in the circle and in the elL'pfis would be .
equal to each other. Thus, I fay, it would be,if fg was to ceaa fY
ce xfY
to cY, that is, if fg was equal to — ^ — . But becaufe of the
fiimilar triangles fgpi oep, % is to ce as fp to cp; and theie<
fSO matbbmatica'l ipsiNciPLts Book III.
fore fg is equal to * j and therefore the angle which
fg fubteods in faA is to the former angle which FG fubtends,
that is to fay^ the motion of the nodes in the ellipfis is to the
ce X fp
motion of the fame in the circle as this fg or ^ to the
** cp
ce X fY
former fgor ^ — $ that is^ as fp X cY to fY x cp^ or as
fp to fY, and c Y to cp ; that is, if ph parallel to TN meet
FP in h, as Fh to FY and FY to FP; that is, as Fh to FPor
Dp to DP, and therefore as the area Dpmd to the area
DPMd. And, therefore, feeing (by corol. 1, prop. 30) the
latter area and AZ* conjun6lly are proportional to the horary
motion of the nodes in Uie circle, the former area and AZ*
oonjun6Uy will be proportional to the horary motion of the
nodes in the ellipfis. Q.E.D.
CoK. Since, therefore, in any given pofition of the nodes,
the fum of all the areas pDdm, in the time while the moon
is carried from the quadrature to any place m, is the area
mpQEd terminated at the tangent of the ellipfis QE ; and the
fum of all thofe areas, in one entire revolution, is the area of
the whole ellipfis; the mean motion of the nodes in the ellipfis
will be to the mean motion of the nodes in the circle as the
ellipfis to the circle ; that is, as Ta to TA, or 69 to 70. And,
therefore, iince (by coroL 2, prop. SO) the mean horary mo-
tion of the nodes in tlie circle is to 16" 35'" 16*^ 36'. as AZ*
to AT% if we take the angle 16" 21'" 3*\ 30\ to the angle
16" So" 16*\ 36\ as 69 to 70, the mean horary motion of the
nodes in the ellipfis will be to 16" 21'" 2>\ 30\ as AZ* to AT* ;
that is, as the fquare of the fine of the diflance of the node
from the fun to the fquare of the radius.
But the moon, by a radius drawn to tfie earth, defcribes
the area in the fyzygies with a greater velocity than it does
that in the quadratures, and upon that account the time is
(M>ntra6led in the fyzygies, and prolonged in the quadratures;
and together with the time the motion of the nodes is like-
wife augmented or diminifhed. But the moment of the area
in the quadrature of the moon was to the moment thereof in
the fyzygies as 10973 to 1 1073 ; and therefore the mean mo-
'n the odiants is to the excefs in the fyzygies^ and to the
Book JIL OF NATURAL PHILOSOPHY. ^21
defe6l.i(i tl^e quadratures^ as 11023^ tbe half fumofthofe
numbers, to their half difference 50. Wherefore filler the
time of the moon's mora in the feveral little equal parts, of its
orbit is reciprocally as its velocity^ the mean time in the oc-
tants will be to tbeexcefs of the time in the quadratures^ and
to the defeiSi of the time in th^ fyzygies arifing from this
caufcj nearly as 1 1023 to 50. But^ reckoning from the qua-
dratures ;tQ the fyzygies^ I find that the excels of the moments
of tlie area, in the feveral places above the leail moment in
tbe quadratures, is nearly as the fquaVe of the fine of' the
moon's diftance from the quadratures ; and therefore the difr
ference betwixt the moment in any place, apd the bfiean mo*»
mentin the o<9;ants, is as the difference betwixt the fquare pf
the fine of the moon's diftance/rom the quadratures, and the
fquare of the fine of 45 degrees, or half the fquare of the ra-
dius; and the increment of the time in the feveral places be-
tween the oiSlants and quadratures, and the decrement thereof
between the o6lants and fyzygies, is in the fame proportion.
But the motion of the nodes, while the moon defer] bes the fe«
veral little equal parts of its orbit, is accelerated or retarded
in tbe duplicate proportion of the time ; for that motion, while
the moon defcribes PM, is (cateris paribus) as ML, and IV^Ii
is in the duplicate proportion of the time. Wherefore .tlie
^notion of the nodes in the fyzygies, in the time while the
moon defcribes given little parts of its orbit, is diminifhed iqi
the duplicate proportion of the number 1 1073 to the number
.1 1023 ; and the decrement is to the remaining motion a^ lOQ
to 10973 ; but to the whole motion as 100 to 11073 nearly.
But tbe decrement in the places between the o6lants and iyzyr
gies, and the increment in the places between the o6lant$
and quadratures, is to this decrement nearly as the whole iq4>7
tion in thefe places to the whole motion in the fyzygies^ and
the difference betwixt the fquare of the fine of the nation's
jdifi;ance from the quadrature^ and the half fquare of the ra-
dius, to the half fquare of the radius conj^un<5lly. Where-
fore, if the nodes are in the quadratures, and we take two
places,, one on one fide, one on the other, equally di^nt
from the odlant sbd other two difi;ant by the fame interval.
222 MAThSMATICAL FBtNCrpLCS Book TH,
one from the fyzyg7> tbe other from the qaadratare^ and
firom the decrements of the motions tn the two pliices be*
tween the fyzygy and odlant we fubtraA the increments of
the motions in the two other places between the o<^nt and
the quadrature, the remaining decrement will be equal to
the decrement in the fyzygy> as will eafily appear by compile*
tatbn ; and therefore the mean decrement^ which ought to
• be fubdu6led from the mean motion of the nodes^ is the fourth
]Murt of the decrement in the fyzygy* The whole horary mo-
tion of the nodes in the fyzygies (when the moon by a ladiiis
drawn to the earth was fuppofed to defcribe an area propor-
tional to the time) was 32" 42"' 1^. And we have fiiewa
that the decrement of the motion of the nodes^ in the time
while the moon^ now moving^with greater velocity^ delcrifoes
the fame fpace> was to this 'motion as 100 to 11073; and
therefore this decrement is 17'" 43*^. IT. The fourth part of
which 4"' 25^"^. 48''. fubtraded from the mean horary motion
•boTe found, 16" 21"' 3*^ 30\ leaves l6" Iff" 37*^. 42*. their
conredl mean horary motion.
If the nodes are without the quadratures, and two places
are confidered, one on one fide, one on the other, equally
diftant from the fyzygies, the fum of the motions of the nodes,
when the moon is in tbofe places, will be to the fum of their
motions, when the moon is in the fame places and the nodes
in the quadratures, as AZ* to AT*. And the decrements of
^e motions arifing from the caufes but now explained will
be mutually as the motions themfelvcs, and therefore the re-
maining motions will be mutually betwixt themfelve^as AZ^
to AT*; and the mean motions will be as the remaining mo^
(ions. And, therefore, in any given pofition of the nodes;,
their correA mean horary motion is to l6" Iff" S?*"". 42''. as
AZ* to AT* ; that is, as the fquare of the fine of the difiance
of the nodes firom the fyzygies to the fquare of the radius.
PROPOSITION XXXIII. PROBLEM XIII.
To find the mean motion of the nodes of the moon. (PI.
12, Fig. 2.)
TTiC yearly mean motion is the fum of all the mean horary
motioBs throughout the courfe of the year. Suppofe that the
jftooft lit. OF NATURAL PHILOSOPHY*. £2Sv
node is HI' N^ and that^ after every hour is elapfed, it isdrawa
back again to its former place ; fo that^ notwithftanding its
proper motion^ it may conflaatly remain in the fame fitua-
iion with refpe6): to the fixed ftars; while in the mean time
the fun S^ hy the motion of the earthy is feen to leave the
node^ and to proceed till it completes its apparent annual
courfb by an' uniform motion. * Let Aa reprefent a given leaft
are, whic4) the right line TS always drawn to the fun, by its
interfedion with the circle NAn, defcribes in the leaft giveik
moment of thne ; and the mean horary motion (from what
we have above (hewn) will be as AZ% that is (becaufe AZ
and ZY are proportional), as the redangle of AZ into ZY,
that is^ as the area AZYa ; and the fum of all the mean ho-
rary motions froin the beginning will be as the fum of alt ih^
areas aYZA, that is, as the area NAZ. But the greateft
AZYa is equal to the re6lan^e of the arc Aa into the radios
of the cirde; and therefore the- fum of all thefe re<S^angIes in
the whole circle will be to the like fum of all the greateft
re^hingles as^ the area of the whole circle to the re^angleof
the whole circumference into the radius, that is, as 1 to ^
Iftat t})e horary motion correfponding to that greateft re^lan^
wa^ 16* Iff^ 37*^. 42^. and this motion in the complete courfe
of the fidereal year, 365*.^. 9', amounts to SQ"* 38' 7"5Gf", and
thettefore the half thereof, 19° 49' 3" 55'", is the mean ma^*
tiott of the nodes correfponding to the whole circle* Atid
the ikit[>t{on of the nodes, in the time while the fun is carried
from N to A, is to 19"* 49' 3" 55'" os the area NAZ to the
whole circle.
llids it would be if the node was after every hour drawii
back again to its former place, that fo> after a complete re^
volution, the fun at the year's end would be fouad again m
the feme node which it had left when the year begun. But,
bebailfe of 'the motion of the node in the mean time, the fun
muft needs meet the nod^ fooner ; and now it remains that we
compdte the abbreviation of the time. I^nce, tten, the,fun,
in the courfeof the year, traviels 300 degrees, an^ the node in
the fame time by its greateft motion would be carried 39? SS'
T 50"V or 39^6355 degrees.; atid the mean motioi^ ^f the node
€24 MATHEMATICAL PRIKCIPLB8 Book lEL
10 any place N is to its mean motion in its quadratures as AZ*
to AT* ; the motion of the fun will be to Uie motion of the
node in N as S60AT' to 39 fiS55 AZ"^ ; that is, as 9/)827646Al>
to AZ*. Wherefore if we fuppofe the circumference NAa
of the whole circle to be divided into little equal parts,
fuch as Aa, the time in which the fun would defcribe the litUe
arc Aa, if the circle was quiefcenti will be to the time of which
it would defcribe the fame arc^ fuppofing the circle together
with tiie nodes to be revolved about the centre T, reciprocallj
as 9,a827646A'P to 9,0827(>46AT +. AZ* ; for the time ig
.reciprocally as tlie velocity with which the little arc is de-
(cribed, and this velocity is the fum of the velocities of both
fun and node. If^ therefore^ the fe<Aor NTA reprefent the
time in which the fun by itfelf, without the motion of the
node^ would defcribe the arc NA, and the indefinitely fmall
part ATa of the fedor reprefent the little moment of the time
in which it would defcribe the leaft arc Aa ; and (letting fall
qY perpendicular upon Nn) if in AZ we take dZ of fuch
length that the redangle of dZ into Z Y may be to the leaft
part ATa of the fedior as AZ* to 9,0827646AT* + AZ*, that
is to fay, that dZ may be to {AZ as AT* to 9,0827646AT^
+ AZ* ; the re6langle of dZ into ZY will reprefent the decre-
ment of the time arifing from the motion of the node, while
the arc Aa is defcribed ; and if the curve NdGn is the locus
where the point d is always found, the curvilinear area NdZ
will be as the whole decrement of time while the whole arc
NA is defcribed ; and, therefore, the excefs of the feSor NAT
above the area NdZ will be as the whole time. But becaofe
the motion of the node in a lefs time is lefs in proportion of
the time, the area AaY? mufl alfo be diminiflied io the lame
proportion ; which may be done by taking in AZ the line eZ
of fuch length, that it may be to die length of AZ a^ A2* to
9,08«7646AT* + AZ* ; for fo the redangle of eZ into ZY
will be to the area AZYa as the decrement of the time ia
which the arc Aa is defcribed to the whole time in which it
would have been defcribed, if the node had been quiefcent ;
and, therefore, tliat re^ngle will be as the decrement of the
motion of the node. And if the curve NeFn is the looas of
Book HI. OF NATIXHAL PHILOSOPHX. 2£5
tljie point e, the whole area NeZ, which is the fum of. all the
deci'ements of that motion, will be as the whole decrement
thereof during the time in which the arc AN is defcribed ;
and the remaining area NAe will be as the remaining motion,
which is the true motion of the node, during the time in
which the whole arc NA is defcribed by the joint motions of
both fun* and node. Now the area of the femi-circle is to the
area of the figure NeFn found by the method of infinite feries
nearly as 793 to 60. But the motion correfponding or propor'-
tional to the whole circle was IQ'' 49' S" 5b'" ; and therefor^
the motion correfp6nding to double the figure NeFn is i° £9'
58" 2'", which taken from the former motion leaves 18° 19^ 3''
53'", the whole motion of the node with refpeiS; to the fixed
ftars in the interval between two of its conjundiions with the
fun ; and this motion fubdu(^ed from the annual motion of
the fun 360^ leaves 34 1"* 40' 54" 1'% the motion of the f^u in
the interval between the fame conjundions. But as this mo-
tion is to the annual motion 360°, fo is the motion of the node
but juft now found 18°' 19' 5" oS'" to its annual motion, which
will therefore be 19^ IB' 1". 23'"; and this is the mean motion
of the nodes in the fidereal year. By afl^ronomical tables, it
is ig"" 21 -21" 50'". The difference is lefs than ^ part of the
whole motion, and feems to arife from the eccentricity of the
moon's orbit, and its inclination to the plane of the ecliptic.
By the eccentricity of this orbit the motion of the nodes is
too much accelerated ; and, on the other hand, by the inch*
nation of the orbit, the motion of the nodes is fomething r^
tarded, and reduced to its jufl; velocity.
PROPOSITION XXX III. PROBLEM XIV.
To find the true motion of the nodes of the moon, (PI. 12^
Fig. 3.) ^
In the time which is as the area NTA —7: NdZ (in the pre**
ceding Fig.) that motion* is as the area NAe, and is thence
given ; but becaufe the calculus is too difficult, it will be bet*
ter to jufe the following conftru^ion of the problem. Aboat
the centre C, with any interval CD, defcribe the circl^
BEFD ; produce DC (o A. lb as AB may be to AC as the
mean motion to half the mean true motiou wti^n the Jpode^
286 MATHEMATICAL PRINClPlEflt Book IIT.
are in their quadratures (that is, as 19^ 18' 1" 23"' to 19^ 4J>'
3" Sa*"', and therefore BC to AC as the difference of thofe
motions 0^ 31' 2" S2"' to the latter motion ig^ 49' 3" M'",
that is, as I to 38i^). Then through the point D
draw the indefinite line Gg, touching the circle in D ; and
if we ;take the angle BC£, or BCF, equal to the douhle
diftance of the fun from the place of the node, as found
by the mean motion, and drawing AE or AF cutting
the perpendicular D6 in G, we take another angle which
Ihall be to the whole motion of the node in the inter-
vai between its fyzygies (that is, to 9* 1 1' 3") as the tangent
DG to the whole circumference of the circle B£D, and add
ifhis lajl angle (for which the angle DAG may be ufed) to the
/mean motion of the nodes, while they are pafling from the
quadratures to the fyzygies, and fubtradl it from their mean
motion while they are paffing from the fyzygies to the qua«
dratures, we ihall have their true motion ; for the true motion
fo found will nearly agree with the true motion which comes
out from affuming the times as the area NTA — NdZ, and
the motion of the jiode as the area NAe ; as whoever will pleafe
to examine and make the computations will find : and this is
the femi-menftrual equation of the motion of the nodes. But
there is alfo a menftrual equation, but which is by no means
neceffary for finding of the moon's latitude ; for fince the
variation of the inclination of the moon's orbit to the plane
of the ecliptic is liable to a twofold inequality, the one femi-
menftrual, the other menftrual, the menftrual inequality of
this variation, and the menftrual equation of the nodes^ fo
moderate and corredl each other^ that in computing the
latitude of the moon bolh may be negleAed.
Cor. From this and the preceding prop, it appears that
the nodes are quiefcent in their fyzygies, but regreffive in
their quadratures, by an hourly motion of I6" I9'" Q^". ; and
that the equation of the motion of the nodes in the odlants is
1** 30' ; all which exsL&\y agree with the phsenomena of the
heavens.
SCHOLIUM.
Mr. Maehin, Aftron., Prof. Grefli., and Dr. Hmry Pember^
ton, feparately fpundout the motion gf the nodes by a'difl€i;&nt
^OA; III. OF NATURAL PHILOSOPRT. $97
method. Mention has been made of this method in another
place. Their feveral papers^ both of which I have feen^ con-
tained two propoiitions^ and exa<S);ly agreed with each other
in both of tbem. Mr. Machiri^ paper coming firft to my
hands^ I (hall here jinfert it.
OF THE MOTION OP THE MOOM's NODES.
^ <' PROPOSITION I.
^ Tbt mean motion of the fun from the node is defined by a
geometric mean proportional between the mean motion of
the fun and that mean motion with which the fun rtcede^
with the greateft fwiftmfi from the node in the qua*,
^^ JLratures.
'^ Let T (PI. 13, Fig. 1) be the earth's place, Nn the line of
*^ the moon's nodes at any given time^ KTM a perpendicular
*' thereto, TA a right line revolving about the ceptre with the
^ '^ fame angular velocity with which the fun and the node
'^ recede from one another, in fuch fort that the angle be*
" tween the cj^uiefcent right line Nn and the revolving line
'^ TA may be always equal to the diftance of the places
^' of the fun and node. Now if any right line TK be
^^ 4ivided into parts TS and SK^ and thpfe parts be taken,
*^ as the mean horary motion of the fun to the mean horary
*' motion of the node in the quadratures^ and there be takea
" the right line TH, a mean proportional between the part
>^ TS and the whole TK^ this right line will be proportional^
^^ to die fun's mean motion from the node«
'^ For let there be defcribed the circle NKpM from t^e
'' centre T and with the radius TK, and about the fame cen-
^' tre, with the femi-axes TH and TN, let there be defcribed
^^ an elUpiis NHnL ; and in the time in which the fun recedeg
- *' from the node through the arc Na, if there be drawn the
f ' right line Tba, the area of the fefikor NTa will be the expo-
• K nent of the fum of the motions of the fun and node in, the
/' fame time, Let^ therefore, the extremely. fmall arc aA b^
'^ that which the right line Tba, revolving according te th^
<^ abovefaid law, will uniformly defcribe in a given particle.
^ of time, and the extremely ihiall feAor TAa will l^ as th^
^ ivifo, of the velocities with which tbip $m w4 A^de ar(9 car*
u
u
9M MATHEMATICAL PRINCIPLES' Book HI.
'^ ried two different ways in that time. Now the fun'is velocity
is almoft uniform, its inequality being fo imall as fcarcely
to produce the leail inequality in the mean motion of the
'' nodes. The other part of this fum, namely, the mean
^' quantity of the velocity of the node, is increafed in the re-
'' cefs from the fyzygies in a duplicate ratio of the fine of its
'^ diftance from the fun (by cor. prop. dl> of this book),, and,
*^ being greateft in its quadratures with the fun in K, is in the
*' fame ratio to the fun's velocity as SK to TS, that is, as (the
^ difference of the fquares of TK and TH, or) the rectangle
'' KHM to TH*. But the ellipfis NBH divides the feAor
'' ATa, the exponent of the fums of thefe two velocities^ into
'^ two parts ABba and BTb, proportional to the velocities.
'* For produce BT to the circle in /5, and from the point B let
'' fall upon the greater axis the perpendicular BG, which be-
*^ ing produced both ways may meet the circle in the points F
'' and f ; and becaufe the fpace ABba is to thefe6lorTBb as
^' the reAangle AB/) to BT* (that redlangle bang equal to the
*' difference of the fquares of TA and TB, becaufe the right
line A/3 is equally cut in T, and unequally in B), therefore
'^ when the fpace ABba is the greateft of all in K, this mtio
" will be the fame as the ratio of the re6);angrle KHM to HT*.
*' But the greateft mean velocity of the node was fhewn above
'^ to be in that very ratio to the velocity of the fun ; and
^'. therefore in the quadratures the fedlor ATa is divided into
^' parts proportional to the velocities. And becaufe the re<5i-
" angle KHM is to HT* as FBf to BG*, and the rectangle
'' AB/I is equal to the redlangle FBf, therefore the little area
*' ABba, where it is greateft, is to the remaining fedior TBb
as the re<Slangle AB/5 to BG*. But the ratio of thefe little
areas always was as the redlangle AB/5 to BT*; and there-
'^ fore the little area ABba in the place A is lefs than its
^' correfpondent little area in the quadratures in the duplicate
'^ ratio of BG to BT, that is, in the duplicate ratio of the fine
^^ of the fun's diftance frona the node. And therefore the
" fum of all the little areas ABba, to wit, the fpace ABN,
*' will be as the motion of the node in the time in which the
^' fun bath been going over the arc ^A fince he left the node;
u
<€
€€
€€
Bdoi IIL OF NATURAL PHILOSOPHY. ^29
*^ and the remaining fpace^ namely, the elliptic fe6lor NTB,
^ will be as the fun's mean motion in the fame time. And
^^ becauie the mean annual motion of the node is that mo-
^' tion which it performs in the time that the fun completes
'* one period of its courfe, the mean motion of the node from
*' the fun will be to the mean motion of the fun itfelf as the
*' area of the circle to the area of the ellipfis ; that is, as the
right line TK to the right line TH, which is a mean pro-
portional between TK and TS ; or, which comes to the
fame^ as the mean proportional TH to the right line TS.
'' PROPOSITION II.
*' The mean motion of the moon's nod<^& being given, to fini
\ " thtir true motion. -
" Let the angle A be the diftance oi the fun from the
mean place of the node, or the fun's mean motion from the
^' node. Then if we take the angle B, whofe tangent is to
the tangent of the angle A as TH to TK, that is, in the fub-
duplicate ratio of the mean horary motion of the fun to the
*^ mean horary motion of the fun from the node, when the
node is in the quadrature, that angle B will be the diftance
of the fun from the node's true place. For join JFT, and,
*^ by the demonftration of the laft proportion, the angle FTN
will be the diftance of the fun from the mean place of the
node, and the' angle ATN the diftiance from the true place,
<^ and the tangents of thefe angles are between themfelves aa
«TKtoTH.
^' CoR. Hence the atigle FTA is the equation of the
•' moon's nodes; and the fine of this angle, where it isgreateft
^ in the p<ftants, is to the radius as KH to TK + TH. But
*' the fine of this equation in any other place A is to the
^ greateft fine as the fine of the fums of the angles FTN +
^' ATN to the radius ; that is, nearly as the fine of double
.^' the diftance of the fun from the mean place of the node
^' (namely, 2FTN) to the radius.
'' SCHOLIUM.
^' If the mean horary motion of the nodes in the quadra^
tures be l6" iG" 37*'. 42^. that is, in a whole fidereal year,
Sff" 38' 7" 60"', TH will be to TK in the fubduplicate ratio
Q 3
xc
it
€€
tt
€€
%90 MATHlBltATlCAL VltlirClPLBS JBook IR,
^ of the number 9^0827646 Uf the number 10^827646, that
'' is, as 18,6524761 to 19,6524761. And, therefore, TH h
^ to HK as 18,6524761 to 1 ; Aat is, as the motion of the
^ fmi in a fidereal year to the mean motion of the node 19*
^ 1^ I" 23f ".
^ But if the mean motion of the moon's nodes in 20 JnKati
^ years is 386^ 5€f 15", as is coUeAed from the obfarvation»
^ made nfe of in the theory of the moon, the mean motion
** of the nodes in one fidereal year will be ig'^ 20* 3 J" 58'".
•' And TH will be to HK as 36(f to 19° 2€f 31" 58" ; that is^
" as 18,61214 to 1: and from hence the mean horary motion
^ of the nodes in the quadratures will come out l6" 18"' 48^""^^
^ And the greateft equation of the nodes in the odants will
•^ be 1** 29' 57".''
PROPOSITION XXXIV. PROBLEM XV.
Tojmd the horary variation of the inclinatiom of the moowl^
orbit to the plane of the ecliptic.
Let A and a (PI. 13, Fig. 2)reprefent the fysygies ; Q and
q the qnadfatures; N and n the nodes; P the place of th«
moon in its orbit ; p the orthographic projection of that plac«
Upon the plane of the ecliptic ; and mTl the momentaneoos
motion of the nod^s as above. If upon Tm we let fall ths
perpendicular PG, and joining pG we produce it till it meei
Tl in g, and join alfo Pg, the angle PGp will be the inclf-
nation of the moon's orbit to the plane of the ecliptic when
the moon is in P ; and the angle Pgp will be the inclination
4>f the fame after ^ fmall moment of time is elapfed; and
therefore the angle GPg will be the momentaneous variation
of the inclination. But this angle GPg is to the angle GTg
•as TG to PG and Pp to PG conjundly. And, therefore, if
for tlie moment of time we afTume an hour, fince the angle
©Tg (by prop. SO) is to the angle 33" 10"' SS^\ as IT X PG
X AZ to AP, the angle GPg <or the horary variation of tfa«i
inclination) will be to the angle S3" 10" 33*% as IT X AZ
X TG X ^ to AT^. Q.E.I.
And i\kvts it would be if the moon was uniformly revolved i^
# cirpular orbit. But if Ibe orbit is ellipticals th^ mean mor
Book IIL OF NATUEAL PHILOSOPHY. 231
tion of tbe nodes will be diminifhed in proportion of the leiTer
axis to the greatei*^ as we have (hewn above ; and the va-
riation of the inclination will be alfo diminifhed in the fame
propcM'tion.
Cob. 1. Upon Nn ere6l the perpendicular TF, and let pM
be the horary motion of the moon in the plane of the ecliptic;
upon QT let fall the perpendiculars pK, Mk, and produce
them till they meet TF in H and H ; then IT will be to AT
as Kk to Mp ; and TG to Hp as TZ to AT ; and, therefoie,
™ r«^ .« t^ 1 Kk X Hp X TZ ^ .
IT X TG will be equal to ?np y that is, equal
TZ
to the area HpMli multiplied into the ratio r— - ; and diere*
fore the horary variation of the inclination will be to 33" IQf^
3S*\ as the area HpMh multiplied into AZ x :jrr- x ~ to
Cor. 2. And, therefore, if the earth and nodes were after
every hour drawn back from their new and inftantly reftored
to their old places, fo as their iituation might continue given
for a whole periodic month together, the whole variation of
the inclination during that month would be to 33" 10'" SS^"".
as the aggregite of all the areas HpMh, generated in the
time of one revolution of the point p (with due regard in
fumming to their proper figns + and — ), multiplied into AZ
X TZ X ^ to Mp X AT' ; that is, as the whole circle
QAqa multipITed into AZ x TZ X p^ to Mp X AT', that
is, as the circumference QAqa multiplied into AZ x TZ x
j^ to 2Mp X AT*.
Cor. 3. And, therefore, in a given pofitioa of the nodes,
the mean horary variation, from which, if uniformly con-
tinued through the whole month, that menftrual variation
might be generated, is to 33" 10'" S3^\ as AZ X TZ x,
|;^tg2AT^ or as Pp x lAt ^^^^ ><4AT5 thatis
Q4
fiSS MATHEMATICAL PRINC1PLB» Book III,
(becaufe Pp is to P6 as the fine of the aforefaid inclination
AZ X TZ
to the radiusj and — - .^ ■ to 4AT as the fine of double the
angle ATn to four times the radius), as the fine of the fame
inclination multiplied into the fine of double the diftance
of the nodes from the fun to four times the fquare of the
radhis. »
Cor. 4. Seeing the horary variation of the inclination, when
the nodes are in the quadratures, is (by this prop.) to the
ungle 33" 10'" 33^\ as IT x AZ x TG X pg to AT, tbat
IT X TG Pp ^^ , . . ^ ^^ ,,
IS, as — YTrf — ^ p7^ to 2 AT, that is, as the fine of double
the diftance of the moon from the quadratures multiplied into
Pp
^t; to twice the radius, the fum of all the horary variations
during the time that the moon, in this fituation of the nodes,
pafles from the quadrature to the fyzygy (that is, in the fpace
J>f ITT-J- hours) will be to the fum of as many angles 33" 10"'
35*''. or 5878", as the fum of all the fines of double the dift-
Pp
ance of the moon from the quadratures multiplied into ^5^
to the fum of as many diameters ; that is, tt the diameter
. . . Pp
nmltiplied into pp to the circumference ; that is, if the in-
cUnation be 6° T, as 7 X tStot to 22, or as 278 to 10000.
And,' therefore, the whole variation, conipofed out of the
fum of all the horary variations in the afore&id time, is
163", or 2' 43".
PROPOSITION XXXV. PROBLEM XVI.
To a given time to find the inclination of the moon^s orbit to
the plane of the ecliptic.
Let AD (PL 14, Fig. J) be the fine of the greatefi; incli-
«iation, and AB the fine of the leaft. Bife6l BD in C ; and
round the centre C, with the interval BC, defcribe the circle
BGD. In AC take CE in the fame proportion to EB as EB
to twice BA. And if to the time given we fet off the angle
AJBG equdl to double the diftance of the nodes from the qui^
jP/aAf^XmJWil .
#»•
T J^
«
*
«H
Book III. OF NATURAL pniLOsoraT. 2dS
driatures^ and npon AD let fall the perpendicular GHj AH-
will be the line of the inclination required. :>
For GE* is equal to GH» + HE* = BHD + HE* =
HBD + HE* — BH* = HBD + BE* — 2BH x BE =
BE* + 2EC X BH = 2EC x AB + SfEC x BH = 2EC
X AH ; wherefore fince 2EC is given, GE* will be as AH.
Now let AEg reprefent double the diftance of the nodes from
the quadratures, in a given 4noment of time after, and the
arc 6g, on account of the given angle GEg, will be as the
diftance 6E, Bui Hh is to Gg as GH to GC, and, there-
fore, Hh is as the reiaangle GH x Gg, ox GH x GE, that
r^w CH
is, as jrp X GE% or -^ x AH^ that is, as AH and the
fine of the' angle AEG conjundlly. If, therefore, in any one
cafe, AH be the fine of inclination, it will increafje by the
fame increments as the fine of inclination doth„ .by cor. S of
the preceding prop, and therefore will always continue equal
to that fiqe. But when the point G falls upon either point
B or D, AH is equal toihisfijie, and therefore remains always
equal thereto. Q.E.D. -
In this demonftration I have fuppofed thatthe.angle BEG^
reprefenting double the diftance of the nodes from the qua^-
dratures, increafeth uniformly ; for I cannot defcend to every
minute circumftance of inequality. Now fuppofe that BEG
is a right angle, and that Gg is in this cafe the horary incre^
ment of double the diftance of the nodes from the fun ; then*,
by cor. 3 of the laft prop, the horary variation of the indinar
tion in the fame cafe will be to S3" 10"' ^^l^ as the rediangle
of AH, the fine of the inclination, into the fine of the right
angle BEG, double the diftance of the nodes irom the fun, t6
four times the fquare of the radius ; that is, as AH, the fine of
the mean inclination, to four times the radius; that is, feeing
the mean inclination is about 5^ 8|, as its finq 896 to 40000,
the quadruple of the radius, or as 29A to 10000. But the
whole variation correfponding to BD, the difference of the
fines, is to this horary variation as the diameter BD to the arc
Gg, that is, conjun6lly as the diameter BD to the femi-cir-
cumference BGD, and as tte time of 2079i%: Hours, in which
flS4 MATHEMATICAL MtlNCIPLSr Bo(A lUf
the node proceeds from the quadratureft to ilie fyzygtet^ to
one honr^ that is^ as 7 to 11, and 2079^ to 1. Wherefore^
compounding all thefe proportions, we (hall have the whole
mriation BD to 33" 10'" S3*\ as 224 x 7 .X 2079^10
1 10000, that is, as 29645 to 1000 ; and from thence tlmt ¥»•
ristion BD will come out l6' 23|". *
. And this is the greateft variation of the inclinatiott, ab-
firadting from the fituation of the moon in its orbit; for if die
nodes are in the fyzygies, the ioclination fnflSers no change
from the various poiitions of the moon. But if the nodes are
in the quadratures, the inclination is lefs when llie modn is in
the fyzygies than when it is in the quadratures by a diflerence
of 2' 43", as we (hewed in cor. 4 of the preceding pirop, ; and
the whole mean variation BD, dinuni(hed by 1' 2 If", the half
t>f this excefs, becomes 15' 2", when the moon is in the qua*
dratures ; and incrcafed by the fame, becomes 17' 45"^ when
the moon is in the (yzygies. If, therefore, the moon be in
the (yzygies, the whole variation in the pa(rage of tiie nodsa
irom the quadratures to the fyzygies will be 17' 45" ; and,
therefore, if the inclination be 5° 17' 20", whtn the nodes are
jtt the fyzygies, it will be 4^ 59' 35" when the nodes are in the
quadratures and the moon in the fyzygies. The truth of all
which is confirmed by obfervations.
Now if the inclinatioi:) of the orbit (hould be requiiied when
4he moon is in the fyzygies, and the nodes any where between
them and the quadratures, let AB be to AD as the £ne of
4*^ 59' 35" to the fine of 5° 17' 20", and take the angle AEG
«qual to double the diftance of the nodes (rom the quadrsluves ;
and AH will be the. fine of the inclination defined. To this
inclination of the orbit the inclination of the fame is eqnai,
when the mo6n is yo^diftant from the nodes. In other fitua*
tions of the moon^ this menftrual inequality, to which the
variation of the inclination is obnoxious in the calculus of the
moon's latitude, is balanced, and in a manner took off, by
the menftrual inequality of the motion of the nodes (as we
faid before), and therefore may be negkdied in the compo^
tation of the faid latitude.
JSooJfc ISL o9 n ATURAL ptfiixxio^fnr^ f^6
SCHOLIUM.
Bj the& compotationft of the lunar motions I was willing to
ihew that by the theory of gravity the motions of the mooa
could be calculated fron^ their phyfical caufes. By the iamt
theory I moreover found. that the annual equation of the
mean motion of the moon arifes from the various dilatation
which the orbit of the moon fuffers from the adUon of the ilm,
according to cor. 6, prop. 66^ book 1. The force of this
a6tion is greater in the perigeon fuu^ and dilates the moon's
orbit; in the apogeon fun it is lefs, and permits the orbit to
be again contraded. The moon moves flower in the dilated,
and fafter in the contradled orbit ; and the annual equation^
by which this inequaUty is regulated^ vanifhes in the apogee
and perigee of the fun. In the mean diftance of the fua,
irom the earth it arifes to about IT 50"; in other diftances
of the fun it is proportional to the equation of the fun'a
centre, and is added to the mean motion of the moon, while
the earth is paffing from its aphelion to its perihelion, and,
fubduiSled w^ile the earth is in the oppofile femi-circle,
faking for the radius of the orbis magnus 1000, and 16f for
jthe earth^s eccentricity, this equation, when of the gi-eateft
fuagnitude, by the theory of gravity comes out 1 T 49". But
tlie eccentricity of the earth feems to be fomething greatet^
ftnd with the 'eccentricity this equation will be augmented in
the fame proportion. Suppofe the eccentricity l6f}, and the
great^ equation will be 1 1' 5 1".
Fairer ; I found that the apogee and nodes of the moon
move fiafier in the perihelion of theejarth, where the force of
the fmfs a49bion is greater, than in the aphelion therof, and
that .in the reciprocal triplicate proportion of the earth's
difUnce from the fun; and hence arife annual equations of
tbofe motions proportional to the equation of the fun's centre.
]Mow the nK)tion of the fun is in the reciprocal duplicate pro-
portion of the earth's diflance from the fun; and the.greateft
equation of the centre which this inequality generates is p
56' ^0", correfponding to the above-mentioned eccentrici^
of the fun, l6f).. But if the motion of the fun had been in
t^ ri^i|npo«al triplicate proportion of the diflance, this ^-
<ft36 MJkTRXMATlCAL f&ijicipLes Bwk III.
quality would have generated the greateft equaiiou e* 64/ 30";
•nd therefore the greateft equations which the inequaliiies of
the motions of the moon's apogee and nodes do generate are
§0 9^ 5A! so" as the mean diurnal motion of the moon's apo-
gee and the mean diurnal motion of its nodes are to the mean
diurnal motion of the fun. Whence the greateft equation
of the mean motion of the apogee comes out l^^^S''^ and the
greateft equation of the mean motion of the nodes ^ 24".
The former equation is added^ and the latter fubduSed^ while
4be earth is pafling from its perihelion to its aphelion, and
<x>ntrariwife when the earth is in the oppofite femi-circk.
By the theory of gravity I likewife found that the aAion
of the fun upon the moon is fomething greater when the
Aranfverfe diameter of the moon's orbit pafleth through the
fun than when the fame is perpendicular upon the line which
joins the earth and the fun ; and therefore the moon's orbit
is fomething larger in the former than in the latter cafe.
And hence arifes another equation of the moon's mean
motion, depending upon the fituation of the moon's apogee
in. rcfpe& of the fun, which is in its greateft quantity when
the moon's apogee is in the oAants of the fun, and Taniihes
when .the apogee arrives at the quadratures or fyzygies ; and
it is added to the nTean motion while the moon's apogee is
pafling from the quadrature of the fun to the fyzygy, and
fubdudied while the apogee is pafling from the fyzygy to the
quadrature. This equation, which I fliall call the femi-annu*
al, when greateft in the o6iants of the apogee, arifes to
ai)out 3' 45" y fo far as I could colleA from the phsenomena:
and this is its quantity in the mean diilance of the fan from
the earth. But it is increafed and diminiflied in the recipro*
cal triplicate proportion of the fun's diftance, and therefore
is nearly 3' 34" when that diftance is greateft, and 3' 56"
when leaft. But when the nroon's apogee is without the
o<^ants, it becomes lefs, and is to its greateft quantity as the
•^ne of double the diftance of the moon's apogee from t|ie
neareft fyzygy or quadrature to the radius.
3y the fame theory of gravity, the a6lion of the fun upon
th^ moon is fomething greater when the line of the moon's
B^^k lit OP NATURAL PHILOSOPHY. 2S7
nodes pafTes through the fun than when it is at right angles
with the line which joins the fun and the earth; and henc&
arifes another equ action of the moon's mean motion^ which I
IhilU call the fecond femi-annual ; and this is greateft when
the nodes are in the o6);ants of the fun^ and vanifties whea
they are in the fyzygies or quadratures ; and in other poiitions
of die nodes is proportional to the fine of double the diftance
of either node from the neareft fyzygy or quadrature. And
il is added to the mean motion of the moon^ if the fun is in
^mimdentia, to the node which is neareft to him, and fub-
dufied if in confequentia ;^ and in the o6lants, where it is of
the greateft magnitude^ it arifes to 47^ in the m^an diftance
of the fun from tjie earth, as I find from the theory of gra-
vity. In other diftances of the fun, this equation, greateft in
the odiants of the nodes, is reciprocally as the cube of the
fun's diftance from the earth ; and therefore in the fun's pe-
rigee it comes to about 49'^ and in its apogee to about 45".
By the fame theory of gravity, the moon's apogee goes
forward at the greateft rate when it is either in conjundUon
with or in oppofition to the fun, but in its quadratures with
the fun it goes backward ; and the eccentricity comes, in the
former cafe, to its greateft quantity ; in the latter to its leaft,
by cor. 7, 8, and g, prop. 66, book 1 . And thofe inequalities,
by the corollaries we have named, are very great, and generate
the principal, which I call the femi-annual equation of the
apogee; and this femi-annual equation in its greateft quan-
tity comes to about 1£^ 18", as nearly as I could coliedl from
the phenomena. Our countryman Horrox was the firft who
advanced the theory of the moon's moving in an ellipfis about
the earth placed in its lower focus. Dr. Halhy improved
tlie notion, by putting the centre of the ellipfis in an epicyqte
whofe^ centre is uniformly revolved about the earth ; and
from the motion in this epicycle the mentioned inequalities
in the progrefs and regrefs of the apogee, and in the quan-
tity of eccentricity, do arife. .Suppofe .the mean diftance df
the moon from the earjbh to be divided into lOOOOO parts, And
let T (PI. 14, Fig. £)iicqfurefent. the earthy and TC the n^pon'^
mean eccentricity of J6Q£» fuch parts^ frodoce TQ toB,
t40 MATHEMATICAL PBIITCIPLBS Book HI.
The atniofpherc of the earth to the height of 35 or 40 mil^
refracts tlie fun's light. Tliis refraction fcatters and fpreads
the light over the earth's {hadow ; aod the difiipated light
near the limits of the ihadow dilates the ihadow. Upon
which account) to the diameter of the ihadow^ a^ it comes
out by the parallax, I add 1 or 1^ minute in lunar eclipfes.
But the theory of the moon ought to be examined and prov-
ed from the phsBnomena, firft in the fyzygies^ then in the qua-
dratures, and laft of all in the oclants ; and whoever pleafes to .
undertake the work will find it not amifs to affume the follow-
ing mean motions of the fun and moon at the Royal Obferva*
.tory of Greenwich, to the laft day of December at noon, anno
1700, O.S. viz. The mean motion of the fun }^ 20° 43' 40",
and of its apogee 7° 44' 30" ; the mean motiim of the moon
«r W 21' 00" ; of its apogee, H 8° 20' 00" 5 and of ite afcend-
ing nbde Si 27° 24' 20" ; and the difference of meridians ber
twixt the Obfervatory at Gre€nwich and the Royal Obfervato-
ry at Paris, 0** 9' 20" : but the mean motion of the moon and
of its apogee are not yet obtained with fufficient accuracy.
PROPOSITION XXXVL PROBLEM XVIL
To find the force of the fun to move the fea^
The fun's force ML or PT to difturb the motions of the
moon, was (by prop. 25) in the moon's quadratures^ to the
force of gravity with us, as 1 to 638092,6; and the force TM
— LM or 2PK in the moon's fyzygies is double that quan-
tity. But, defcending to the furface of the earth, thefe forces
arc diminifhed in proportion of the diftances from the cen-
tre of the earth, that is, in the proportion of 60i to 1 ; and
therefore the former force on the earth's furface is to the force
of gravity as 1 to 38604600 ; and by this force the fea is
depreffed in fuch places as are 90 degrees diftant from the
fun. But by the other force, which is twice as great, the fea
is raifed not only in the places dire6lly under the fun, but in
thofe alfo which are diredly oppofed to it ; and the fiim of
thefe forces is to the force of gravity as 1 to 12868200.
And becaufe the fame force excites the fame motion, whe-
ther it depreiTes* the waters in thofe places which are QO der
j^rees diftant from the ItiO;, or raifes them in the places wbicb
Booft in« OP irATURAt PfilLOSOPHT* fit
are diredly tmder anddiredUy oppofed to the ftin^ the afore-*
faid fum will be the total force of the Ain to difturb the fea^
and will have the fame effeSt as if the whole was employed la
raifing the fea in the places dire6):ly under and direcSUy op*
pofed to the fun^ and did not ad); at all in the places which
are QO degrees removed from the fun.
And this is the force of the fun to difturb the fea in any
given place^ where the fun is at the fame time both vertical^
and in its mean diftance from the earth. In other pofitions
of the fnn^ its force to raife the fea is as the verfed fine of
double its altitude above the horizon of the place diredUy^ and
the r'^be of the diftance from the earth reciprocally.
Co .. Since the centrifugal force of the parts of the earthy
arifing from the earth's diurnal motion^ which is to the force
of gravity as 1 to 9.69, raifes the waters under the equator to
a height exceeding that under the poles by 85472 Paris feel^
as above^ in prop* 19^ the force of the fun^ which we have
now fhewed to be to the force of gravity as 1 to 12868200^
and therefore is to that centrifugal force as £89 to 12866200^
or as 1 to 44597, will be able to raife the waters in the places
diredlly under and directly oppofed to the fun to a height ex^
ceeding that in the places which are 90 degrees removed from
the fun only by one Paris fpot and 113-^ inches; for this
meafuie is to the meafure of 85472 feet as 1 to 44527*
PROPOSITION XXXVII. PROBLEM XVIII.
Tojind the force of the moon to move the fea.
The force of the moon to move the fea is to be deduced
from its proportion to the force of the fun^ and this propor*
lion is to be colle&ed from the proportion of the motions of
the fea, which are the effe6b of thofe forces. Before the
mouth of the river Avon, three miles below Brifiol, the height
of the afcent of the water in the vernal and autumnal fyzy-
gies of the luminaries (by the obfervations of Samuel Sturmjf}
amounts to about 45 feet, but in the quadratures to 25 only.
The former of thofe heights arifes from the fum of the afore-
faid forces, the latter from their difference. If^ therefore, S
and L are fuppofed to reprefent refpe6tively the forces of the
iiiQ and moon while they are in the equator^ as well as in
Vol. II. / R
342 MATHEMATICAL PBINCIFLBH Book III.
their mean diftances from the earth, we fliall have L 4- S to
L — S as 45 to C5> or as 9 to 5.
At Plymouth (by the obfervations of Samuel Colcprefa) the
tide in its mean height rifes to about l6 feet, and in the
fpring and autumn the height thereof in the fyzygies may
exceed that in the quadratures by more than 7 or 8 feeU
Suppofe the greatcft difference of thofe heights to be 9 feet,
and L + S will be to L — S as 20{ to 1 1|, or as 41 to 23 ; a
proportion that agrees well enough with the former. But
becaufe of the great tide at Brijlol, we are rather to depend
upon the obfervations of Sturmy ; and, therefore, till we pro-
cure fomething that is more certain, we (hall ufe the propor-
tion of 9 to 5.
But becaufe of the reciprocal motions of the waters, the
greateft tides do not happen at the times of the fyzygies of
the luminaries, but, as we have faid before, are the third in
order after the fyzygies; or (reckoning from the fyzygies)
folbw next after the third appulfe of the moon to the meri-
dian of the place after the fyzygies; or rather (as Sturmy ob-
ferves) are the third after the day of the new or full moon, or
rather nearly after the twelfth hour from the new or full
moon, and therefore fall nearly upon the forty-third hour after
the new or full of the moon. But in this port they fall out
about the feventh hour after the appulfe of the moon to the
meridian of the place ; and therefore follow next after the
appulfe of the moon to the meridian, when the moon is diftant
from the fun, or from oppofition with the fun by about 18 or
19 degrees in confequentia. So the fummer and winter fea-
fons come not to their height in the folftices themfelves, but
when the fun is advanced beyond the folftices by about a
tenth part of its whole courfe, that is, by about 36 or 37 de-
grees. In like manner, the greateft tide is raifed after the ap<
"pulfe of the moon to the meridian of the place, when the
moon has paffed by the fun, or the oppojitioti thereof, by
about a tenth part of the whole motion from one greateft tide
to the ttext following greateft tide. Suppofe that difiance
about ]8i degrees; and the fun's force in this diftanceof the
moon from the fyzygies and quadratures will be of lefs mo-
Hook in. OP NATURAL PHILOSOPHY. 243
ment to augment and diminilh that part of the motion of the
fea which proceeds from the motion of the moon than in the
iyzjgies and quadratures themfelves in the proportion of the
radius to the co-fine of double this diftance^ or of an angl# of
37 degrees ; that is, in proportion of 10000000 to 7986355 ;
and, therefore, in the preceding analogy, in place of S we
muft put 0,79863558.
But farther ; the force of the moon in the quadratures muft
. be diminifhed, on account of its declination from the equa-
tor; for the moon in thofe quadratures, or rather in 18| de-
grees paft the quadratures, declines from the equator by about
22° 13' ; and the force of either luminary to move the fea is
diminifhed as it declines from the equator nearly in the dupli-
cate proportion of the co-fine of the declination ; and there-
fore the force of the moon iii thofe quadratures is only
0,8570327L; whence we have L+0,7986355S to 0,8570327L
— 0,79863558 as 9 to 5.
Farther yet ; the diameters of the orbit in which the moon
fhould move> fetting afide the confideration of eccentricity,
are one to the other as 69 to 70 ; and therefore the moon's
diftance from the earth in the fyzygies is to its diftance in the
quadratures, cateris paribus, as 69 to 70 ; and itl^ difiances^
when I8i degrees advanced beyond the fyzygies, where the
greateft tide was excited,"and when 18| degrees pafled.by the
quadratures, where the leafi tide was produced, are to its mean
diftance as 69,098747 and 69,897345 to 69f. But the force
of the moon to move the fea is in the reciprocal triplicate
proportion of its diftance ; and therefore its forces, in the
greateft and leaftof thofe diftances, are to its force in its
mean diftance ns 0,9830427 and 1,017522 to 1. From
.whence we have 1,017522L x 0,79863558 to 0,9830427 X
0,8570327 L— 0,79863558 as 9 to 5; and S to L as 1 td
4,4815. Wherefore fince the force of the fun is to the force
of gravity as 1 to 12868200, the moon's force will.be to the
force of gravity as I to 287 1400.
Cor. I. Since the waters excited by the fun's force rife to
the height of a foot and 1 1-^ inches, the moon's force will
raife the fame to the height of 8 feet and 7-9^ inches \ and the
R 2
046 MATHEMATICAL PRINCIPLES Book III.
feet, and the mean diftance of the centres of the earth and
moon, conliiling of ()0\ fuch femi-diameiers, is equal to
1187379440 feet. And this diftance (by the preceding cor.)
is to the diftance of the moon's centre from the common cen*
tift of gravity of the earth and moon as 40>788 to 59^788;
which latter diftance, therefore, is 1138^68534 feet. And
jince the moon^ in refpecl of the fixed ftars, performs its re-»
volution in i27<>. 7^. 43^', the verfed fine of that angle which
the moon in a minute of time defcribes is 12752341 to the ra-
dius 1000,000000,000000 ; and as the radius b to this verfed
fine, fo are 1 158268534 feet to 14,7703353 feet. The moon,
therefore, falling towards the earth by that force which re-
tains it in its orbit, would in one minute of time defcribe
14,7706353 feet; and if we augment this force in the pro-
portion of 178H to 177H> we fhall have the total force of
gravity at the orbit of the moon, by cor. prop. 5 ; and the
moon falling by this force, in one minute of time would de.
fcribe 14,8538067 feet. And at the 60th part of the diftance
of the moon from the earth's centre, that is, at the diftance
of 197896573 feet from the centre of the earth, a body falling
by its weight, would, in one fecond of time, likewife defcribe
14,8538067 feet. And, therefore, at the diftance of 19615800,
which compofe one mean femi-diameter of the earth, a heavy
body would defcribe in falling 15,11175, or 15 feet, 1 inch,
and 4Tr lines, in the fame time. This will be the defcent of
bodies in the latitude of 45 degrees. And by the foregoing
table, to be found under prop. 20, the defcent in the latitude
of Paris will be a little greater by an excefs of about y parts
of a line. Therefore, by this computation, heavy bodies in
the latitude of Paris falling in vacuo will defcribe 15 Paris
feet, 1 inch, 4|4 lines, very nearly, in one fecond of time.
And if the gravity be dimini(hed by taking away a quantity
equal to the centrifugal force arifing in that latitude from the
earth's diurnal motion, heavy bodies falling there will de-
fcribe in one fecond of time 15 feet, 1 inch, and l| line.
And with this velocity heavy bodies do really fall in the
latitude of Paris, as we have (hewn above in prop. 4
and 19.
Book III. OF NATUBAL PHILOSOPHY. 247
' Cor. 8. The mean diftance of the centres of the earth and
moon in the fyzygies of the moon is equal to 60 of the greatelt
femi-diameters of the earthy fubdi|6):ing only about one 30th
part of a femi-diameter ; and in the moon's quadratures the
mean diftance of the fame centres is 60|- fuch femi-diameteAr
of the earth ; for thefe two diftances are to the mean diftance
of the moon in the o6iants as 69 and 70 to 6dh ^Y P^op. 28.
6oR. 9. The mean diftance of the centres of the earth and
moon in the fyzygies of the moon is 60 mean femi-diameters
of the earthy and a 10th part of one femi-diameter ; and in
the moon's quadratures the mean diftance of the fame cen«
tresis 6I mean femi-diameters of the earthy fubdudiing one
30th part of one femi-diameter.
Cor. 10. In the moon's fyzygies its mean liorizontal pa«
rallax in the latitudes of 0« 30, 38, 45, 52, 60, gO degrees is 57'
20", 57' 16", 57' 14^ 57' 12", 57' 10", 57' 8", 57' 4", re-
fpedively.
In thefe computations I do not confider the magnetic at*
tra6lion of the earth, whofe quantity is very fmall and un-
known : if this quantity ihould ever be found out, and the
meafures of degrees upon the meridian, the lengths of ifo-
chronous petadulums in different parallels^ the laws of the
motions of the fea, and the moon's parallax, with the appa-
rent diameters of the fun and moon, (hould be more exadly
determined from phenomena : we ihould then be enabled to
bring this calculation to a greater accuracy.
PROPOSITION XXXVII. PROBLEM XIX.
To find the figure of the moon*$ body.
If the moon's body were fluid like our fea, the force of the
earth to raife that fluid in the neareft and remoteft parts
would be to the force of the moon by which our fea israifed
in the places under and oppoiite to the moon as the accele-
rative gravity of the moon towards the earth to the acoele-^
rative gravity of the earth towards the moon, and the dia-
meter of the moon to the diameter of the earth conjun^y ;
that is, as 39^788 to 1, and 100 to 365 conjundily, or as 1081
to 100. Wherefore, fince our fea, by the force of the moon,
is raifed to 8| fe^t, the lunar fluid would be raifed by th^
R4*
!•
ferce of the earth, to 93 feet; and upoa this account the figure
of the moon would be a fpberoid> whofe greateft diam^ier
produced would pais through the centre of the earthy and ex«
ceed the diametera perpendicular thereto by 186 feet. Sodb
ft figure^ therefcNre^ the moon affisda, and muft have put on
from the beginning. Q«£.L
CoR. Hence it is that the fame face of the moon alwaj*
relpeds the earth ; nor can the body of the moon poffibly
reft in any other pofition^ but would return always by a li^
luratory motion to thisfitoation^ but tbofe librations, howeTer>
mnft beexceedingly (low^ becaufe of the weaknefs of the forces
which excite them; fo that the face of the moon, which
fliould be always obverted to the earth> may^ for the reafon
affigned in prop. ITy be turned towards the other focus of the
moon's orbit^ without being inunediateiy drawn back, and
converted again towards the earth.
LEMMA I.
If APEp (PI. 14, Fig. 3) reprefent the earth umforml^
denfe, marked with the centre C, the poles P^ p^ and the
tquator AE ; and if about the centre C, with the radius CP,
wefuppofe thefphere Pape to be dtfcribed, and QR to ds*
note the plane on which a right line, drawn from the centre
of the fun to the centre of the earth, infiftsat right angles;
and farther fuppofe that thejeveral particles of the whole
exterior earth PapAPepE, without the height of the f aid
fphere, endeavour to recede towards this fide and that fide
from the plane QR, every particle by a force proportional
to its difiance from that plane ; I fay^ in thefirfiplaccy that
the whole force and efficacy of all the particles that arefituate
in AE, the circle of the equator, and difpofed uniformly
without the globe, encompaffing the fame after the manner
of a ring, to wheel the earth about its centre, is to the whole
force and efficacy of as many particles in that point A of the
equator which is at the greateft diftancefrom the plane QR,
to wheel the earth about its centre with a like circular mo»
tion, as 1 to 2. And that circular motion will be performed
about an axis lying in the common fe&ion of the equator
Und the plane QR.
Bodk IlL 09 NAltritAL PHILOBOVHT. 149
For let there be defcribed from the centre K^ with the dia-*
meter Il^^the femi-circle INLK. Suppofe the femi-circum*
ference IN L to be divided into innumerable equal parts^ and
from the feveral parts N to the diameter IL let fall the finea
NM« Then the fums of the fqaares of all the fines NM
will be equal to the fums of the fquares of the fines KM^ and
both fnms together will be equal to the fums of the fquares
of as many femi-diameters KN ; and therefore the fom of
the fquares of all the fines NM will be but half fo great
as the fum of the fquares of as many femi-dimneters
KN.
Suppofe now the circumference of the circle A£ to be di-*^
vided into the like nuinber of little equal parts^ and fit)ni
every fuch part F a perpendicular FG to be let fall upon the
plane QR^ as well as the perpendicular AH from the point A^
Then the force by which the particle F recedes from the plane
QR will (by fuppofition) be as that perpendicular F6; and
this force multiplied by the diftance CG will reprefent the
power of the particle F to turn the earth round its centre«r
And, therefore, the power of a particle in the place F will be''
to the power of a particle in the place A as FG x GC to
AH X HC ; that is, as FC^ to AC* : and therefore the whole
power of all tHe particles F^ in their proper places F^ wiU be
to the power of the like number of paiticles in the place A
as the fum of all the FC^ to the fum of all the ACV
that is (by what we have jdemonftrated before), as 1 to 2*
Q.E.D.
And becaufe the adtion of thofe particles is exerted in the
dire<^ion of lines perpendicularly receding from the plane
QR, and that equally from each fide of this plane, they will
wheel about the circumference of the circle of the equator,
together with the adherent body of the earth, round an
axis which lies as well in the plane QR as in that of the
equator,
LEMMA n.
The fame things ftUlfuppo/id, I fajf, in the fecond place, that
the total foru or power of all the particles Jituatcd evert/
t50 MATHBMATICAI. FRINC1VLE9 Boidc IIL
where about thefpkere to turn the earth about the /aid axis
is to the whole force of the like number of particles, uniform-
ly difpofed round the whole circumference of the equator
AE in thefq/hion of a ring, to turn the whole earth about
with the like circular motion, (uQ,to 5. (PI. 14, Fig. 4.) *
For let IK be any lefler circle parallel to the eqaator AE,
and let L, 1 be any two equal particles in this circle, fiUiated
without the fphere Pape; and if upon the plane QR, which
is at right angles with a radius drawn to the fun, we let fall the
perpendiculars* LM, Im, the total forces by which tbefe par-
ticles recede from the plane QR will be proportional to the
perpendiculars LM, Im. Let the right line LI be drawn
parallel to the plane Pape, and bife6t the fame in X; and
through the point X draw Nn parallel to the plane QR, and
meeting the perpendiculars LM, lin, in N and n; and upon
the plane QR let fall the perpendicular XY. And the con-
trary forces of the particles L and 1 to wheel about the earth
contrariwife are as USl x MC, and lin x mC; that is, as.
LN X MC + NM X MC, and In x mC— nm x mC;
or LN X MC + NM X MC, and LN x mC — NM x
mC, and LN x Mm — NM x "^IC + mC, the difference
of the two, is the force of both taken together to turn the
earth round. The affirmative part of this difference LN X
Mm, or 2LN x NX, is to 2A1I x HC, the force of two par-
ticles of the fame fize fituated in A, as LX* to AC* ; and the
negative part NM X MC + mC,or2XY X CY,isto2AH
X HC, the force of the fame two particles fituated in A; as
CX* to AC*. And therefore the difference of the parts, that
is, the force of the two particles L and 1, taken together, to
wheel the earth about, is to the force of two particles, equal
to the former and fituated in the place A, to turn in like man-
ner the earth -round, as LX* — CX* to AC*. But if the cir-
cumference IK of the circle IK is fuppofed to be divided into
an infinite number of little equal parts L, all the LX* will
be to the like number of IX* as 1 to 2 (by lem. 1); and to the
fame number of AC* as IX* to 2AC*; and the fame number
of CX* to as many AC* as SCX* to sAC*. Wherefore the
united forces of all the particles in the circumference of th^
Book in. OP NATURAL PHILOSOPHY. £51
circle IK are to the joint forces of as many particles in the
place A as IX* — SCX* to 2 AC*; and therefore (by lem. 1)
to the united forces of as many particles in the circumference
of Ae Circle AE as IX* — 2CX* to AC*.
Now if Pp, the diameter of the fphere, is conceived to be
divided into an infinite number of equal parts^ upon which
a like number of circles IK are fuppofed to infift^ the matter
in the circumference of every circle IK will be as IX*; and
therefore the force of that matter to turn the earth about will
be as IX* into IX* —^ SCX^ ; and the force of the fame mat-
ter, if it was fituated in the circumference of the circle AE,
would be as IX* into AC*. And therefore the force of all
the particles of the whole matter fituated without the fphere in
the circumferences of all the circles is to the force of the
like number of particles fituated in the circumference of the
greateft circle AE as all the IX* into IX* — 2CX* to as
many IX* into AC*; that is, as all the AC* — CX* into
AC* — 3CX* to as many AC* — CX* into AC*; that is,
as all the AC* — 4AC* X CX* + SCX* to as many AC*—
AC* X CX* ; that is, as the Ivhole fluent quantity, whofe flux-
ion is AC* — 4AC* X CX* + 3CX*, to the wholefluent quan-
tity, whofe fluxion is AC* — AC* x CX* ; and, therefore,
by the method of fluxions, as AC* x CX — |AC* x CX^
+ |CX5 to AC* X CX — ^AC* X CX^ ; that is, if for CX
we write the whole Cp, or AC, as ^AC* to |AC' ; that is, as
2 to 5. Q.E.D.
LEMMA III.
The fame things ftill fuppofed, I fay, in tlte third place, that
the motion of the whole earth about the axis above-named
arifing from the motions of all the particles, will be to the
motion of the aforefaid ring about the fame axis in a pro*
portion compounded of the proportion of the matter in the
earth to the matter in the ring ; and the proportion of three
fquares of the qtuidrantal arc of any circle to twofquares of
its diameter, that is, in the proportion of the matter to the
matter, and of the number ^9,5^1 5 to the number 1000000.
For the motion of a cyhnder revolved about its quiefcent
axis is to the motion of theinfcribed fphere revolved together
with it as any four equal fquares to three circles infcribed iri
t52 MATHEMATICAL fRINCIPtlS Book JSL
three of thofe fquares ; and the motioa of tbit cylinder is to
the motion of an exceedingly thin ring furrounding both
fphere and cylinder in their common contaA as double the
matter in the cylinder to triple the matter in the ring ; and
this motion of the ring, uniformly continued about the axis
of the cylinder, is to the uniform motion of the fame abont
its own diameter performed in the fame periodic time as the
circumference of a circle to double its diameter.
HYPOTHESIS IL
If the other parts of tltt earth were taken amay, and the re-
maining ring was carried alone about the fan in the orbit (^
the earth by the annual motion, while by the diurnal motion
it was in the mean time revolved about its own axis inclined
to the plane of the ecliptic by an angle of 2di degrees, the
motion of the equino&ial points would be the fame, whether
the ring were fluid, or whether it confified (^ a hard umA
rigid matter.
PROPOSITION XXXIX. PROBLEM XX.
To find the preceffion of the equinoxes.
The middle horary motion of the moon's nodes in a circular
orbit, when the nodes are in the quadratures^ was 16" 33"'
l6'^ 36^ ; the half of which, 8" 17'" S8»". 18^ (for thereafons
above explained) is the mean horary motion of the nodes in
fuch an orbit, which motion in a whole fidereal year becomes
£0^ 11'46". Becaufe, therefore, the nodes of the moon in
fuch an orbit would be yearly transfered 9,0"^ 11' 46" in antC'
cedentia ; and, if there were more moons, the motion of the
nodes of every one (by cor. l6, prop, 66, book 1) would be
as its periodic time ; if upon the furface of the earth a moon
was revolved in the time of a fidereal day, the annual motion
of the nodes of this moon would be to 20** 1 1' 46" as 23*». 56',
the fidereal day, to 0,7^. 7**. 43', the periodic time of our moop,
that is, as ]436 to 39343. And the fame thing would happen
to the nodes of a ring of moons encompaffing the earth, whe-
ther thefe moons did not mutually touch each the other, or
whether they were molten, and formed into a continued ring,
or whether that ring Ibould become rigid and inflexible.
3ook III. OP NATURAL PHILOSOPHY, £53
Let U5^ then^ fappofe that this ring is in quantity of mister
equal to the whole exterior earth PapAPepE^ which lies with*
out the fphere Pape (fee fig. lem. 2) ; and becaufe this fphere
is to that exterior earth as aC* to AC* -;— aC*, tliat is (feeing
PC or aC the leaft femi-diameter of the earth is to AC the
greateft femi-diameter of the fame as 229 to 230)> as 52441
to 459 ; if this ring encompaifed the eaarth round the equator^
and both together were revolved about the diameter of die
ring, the motion of the ring (by lem. 3) would be to the mo-
tion of the inner fphere as 459 to 52441 and 1000000 to
925275 conjun<9Jy^ that is^ as 4590 to 485223 ; and there-
fore the motion of the ring would be to the fum of the mo*
tions of both ring and fphere as 4590 to 489813. Where-
fore if the ring adheres to the fphere^ and communicates its
motion to the fphere^ by which its nodes or equinoiSlial points
recede^ the motion remaining in the ring will be to its former
motion as 4590 to 489813; upon which account the motion
of the equinoctial points will be diminiihed in the fame pro-
portion. Wherefore the annual motion of the equinodlial
points of the body^ compofed of both ring and fphere^ will be
to the motion 20° 11' 46" as 1436 to 39343 and 4590 to
489813 conjunfily, that is, as 100 to 292369. But the forces
by which the nodes of a number of moons (as we explained
above), and therefore by which the equino6lial points of the
ring recede (that is, the forces 3lT^ in fig. prop. 30), are in
the feveral particles as the diftances of thofe particles from
ihe plane QR ; and by thefe forces the particles recede from
that plane : and therefor^ (by lem. 2) if the matter of the
ring was fpread all over the furface of the fphere, after the
iafhion of the figure PapAPepE, in order to make up thai
exterior part of the earth, the total force or power of all the
paiticles to wheel about the earth round any diameter of the
^. equator, and therefore to mov^ the equino6lial points, would
become lefs than before in the proportion of 2 to 5. Where-
fore the annual regrefs of the equinoxes now would be to
!810^ 1 1' 46" as 10 to 73092 ; that is, would be 9" 56"' 50*\
But becaufe the plane of the equator is inclined to that of
Ihe ecliptic^ this motion is to be diminiflied in the proportion
<2J4 MATHEMATICAL PRINCIPLM BookTlh
of the fine onoG (which is the co-fine of 23i deg.) to the ra-
dius 100000; and the remaining motion will now be 9* 7**
20'''. which is the annual preceflion of the equinoxes arifing
from the force of the fun.
But the force of the moon to move the fea was to the force
of the fun nearly as 4>481i to 1 ; and the force of the moon
to move the equinoxes is to that of the fun in the fame pro-
|K>rtion. Whence the annual preceflion of the equinoxes pro*
cceding from the force of the moon comes out 40" 59f" 52^,
and the total annual preceflion arifing from the united forces
of both will be 50" 00'" 12*^ the quantity of which motion
agrees with the phaenomena ; for the preceflion of the equi*
noxes, by aftronomical obfervations> is about 50f' yearly.
If the height of the earth at the equator exceeds its height
at the poles by more than 17i miles^ the matter thereof will
be more rare near the furface than at the centre ; and the
preceflion of the equinoxes will be augmented by the excefs
of height^ and diminiflied by the greater rarity.
And now we have defcribed the fyfl;em of the fun, the earth,
moon, and planets, it remains that we add fomething about
the comets,
LEMMA IV.
That the comets are higher than the moon, and in the regions
of the planets.
As the comets were placed by aftronomers above the moon,
becaufe they were found to have no diurnal parallax, fo their
annual parallax is a convincing proof of their defcending into,
the regions of the planets ; for all the comets which move in
a dire6l courfe according to the order of the figns, about the
end of their appearance become more than ordinarily flow or
retrograde, if the earth is between them and the fun ; and
more than ordinarily fwift, if the earth is approaching to a
heliocentric oppofition with them ; whereas, on the other
hand, thofe which move againfl^ the order of the figns, to-
wards the end of their appearance appear fwifter than they
ought to be, if the earth is between them and the fun ; and
flower, and perhaps retrograde, if the earth is in the other
fide of its orbit. And thefe appearances proceed chiefly from
J^tauXiV^WlT.
Jfa^£^4
4F^^
0>
: ^>^ / /
R
K
• ik
Book III. OF NATURAL PHILOSOPHY. 255
the diverfe fituations which the earth acquires in the courfe
of its motion^ after the fame manner as it happens to the pla-
nets^ which appear fometimes retrogrades fometimes more
flowly, and fometimes more fwiftly, progreffive, according aa
the motion of the earth falls in with that of the planet^ or is
direiSted the contrary way. If the earth move the fame way
with the cornet^ but^ by an angular motion about the fun^ fo
much fwifter that right lines drawn from the earth to the
comet converge towards the parts beyond the comet^ the
comet feen from the earth, becaufe of its flower motion, will
appear retrograde ;* and even if the earth is flower than the
comet, the motion of the earth being fubdu6led, the motion
of the comet will at leafl: appear retarded ; but if the earth
tends the contrary way to that of the comet, the motion of
the comet will from thence appear accelerated ; and from
this apparent acceleration, or retardation, or regreflive mo-
tion, the diftance of the comet may be inferred in this man-
ner. LetTQA, TQB, TQC (PL 15, Fig. 1), be three ob-
ferved longitudes of the comet about the time of its firfl^ ap-
pearing, and TQF its lafl: obferved longitude before its difap-
pearing. Draw the right line ABC, whofe parts AB, BC,
intercepted between the right lines QA and QB, QB and QC,
may be one to the other as the two times between the three
firft obfervatious. Produce AC to G, fo as AG may be to
AB as the time between the firfl: and laft obfervation to the
time between the firft and fecond ; and join QG. Now if the
cornet did move uniformly in a right line, and the earth either
flood ftill, or was likewife carried forwards in a right, line by
an uniform motion, the angle TQG would be the longitude of
the comet at the time of the laft obfervation. The angle,
tlierefore, FQG, which is the difference of the longitude, pro-
ceeds from the inequality of the motions of the comet and the
earth; and this angle, if the earth and comet move contrary
ways, is added to the angle tOG, and accelerates the appa-r
rent motion of the comet ; but if the comet move the fame
way with the earth, it is fubtradled, and either retards the
motion of the comet, or perhaps renders it retrograde, as we
have but now explained. Tliis angle, therefore, pr9ceeding
956 MATHEMATICAL FRINCIPLES Book Ul.
chiefly from the motion of the earthy is juftly to be efteemed
the parallax of the comet ; negle6iingy to wit, fome little in*
crement or decrement that may arife from the unequal mo*
tion of the comet in its orbit ; and from this parallax we thus
deduce the diftance of the comet. Let S (PI. ] 5, Fig. 2) re-
prefent the fun, acT the orbis magntis, a the earth's place in
the firft obfervation, c the place of the earth in the thiid ob*
fenration, T the place of the earth in the lall obfervation^ and
TT a right line drawn to the beginning of Aries. Set off the
angle TTV equal to the angle TQF, that is, equal to the Ion*
gitude of the comet at the time when the eardi is la T ; join
ac, and produce it to g, fo as ag may be to ac as AG to AC ;
and g will be the place at which the earth would have ar«
rived in the time of the laft obfervation, if it had continued
to move uniformly in the right line ac. Wherefore^ if we
draw gY parallel to TY^ and make the angle YgV #qual to the
angle YQG, this angle YgV will be equal to the longitude of
the comet feen from the place g, and the angle TVg will be
the parallax which arifes from the earth's being transfered
from the place g into the place T ; and therefore V will be
the place of the comet in the plane of the ecliptic. And this
place V is commonly lower than the orb of Jupiter.
The fame thing may be deduced from the incurvation of
the way of the comets ; for thefe bodies move almoft in great
circles, while their velocity is great ; but about the end of their
courfe, when that part of their apparent motion which arifes
from the parallax bears a greater proportion to their whole
apparent motion, they commonly deviate from thofe circles,
and when the earth goes to one fide, they deviate to the
other ; and this deflexion, becaufe of its correfponding with
the motion of the earth, muft arife chiefly from the parallax;
and the quantity thereof is fo confiderable, as, by my compu*
tation, to place the difappearing comets a good deal lower
than Jupiter. Whence it follows that when they approach
nearer to us in their perigees and perihelions they often de-
fcend below the ov\^ of Mars and the inferior planets.
The near approach of the copiets is farther confirmed front
the light of their beads ; for the light of a celeftial body, ilhi-
Book lUL OP. NATURAL PHILOSOPHY. 2^7
xnioaled, by the fun, and receding to remote parts^ is dimi-
niflied in the quadruplicate proportion of the diftance ; to wit,
in one. duplicate proportion^ on account of the increafe of the
diflancpe from the fun, and in another duplicate proportion^
on account of the decreafe of the apparent diameter. Where-
ibre if both the quantity of light and the apparent diameter of
a comet are given, its diftance will be alfo given, by taking the
diftance of the comet to tjie diftance of a planet in the diretSl
proportion of their diameters and the reciprocal fubduplicate
proportion of -their lights. Thus, in the opmet of the year
16829 Mr. Flamjled obferved with a telefcope of l6 feet, and
ineafured with a micrometer, the leaft diameter of its head,
£' 00; but the nucleus or ftar in the middle of the head
Scarcely ^mounted to the tenth part of this meafure ; and
therefore its diameter was only 1 1" or 12"; but in the light
and fplendor of its head it furpafled that of the comet in the
.year l680, and might be compared with the f^aj^ of the firft
or fecond magnitude. Let us fuppofe that Skttira with its
ling was ^bout four times more lucid ; and becaufe the light
of the ring was almoft equal to the light of the globe within,
and the apparent diameter of the globe is about 21'', and
therefore the united light of both globe and ring would be
equal to the light of a globe whofe diameter is 30", it follow^
jthat the diftance of the comet was to the diftance of Saturn a$
1 to \/4 inverfely, and 12" to 30 diredlly;. that is, as 24 to 30,
or 4 to 5. Again; the comet in the month of April l665, as
Hevelius informs us, excelled almoft all the fixed ftars in
fplendor, and even Saturn itfelf, as being of a much more
vivid colour; for this comet was more lucid than that other
which had appeared about the end of the preceding year, and
bad been compared to the ftars of the firft magnitude. The
diameter of its head was about 6' ; but the nucleus, com-
pared with the planets by means of a telefcope, was plainly
Je& than Jupiter; and fometimes judged lefs, fometimqs
judged equal, to the globe of Saturn within the ring. Since,
then, the diameters of the heads of tlie comets feldom exceed
.8' or 12', and the diameter of the nucleus or central ftar is
but about a tenth or perhaps fifteenth part of the diameter of
Vol. IL S
/
£58 MATHEMATICAL PRINCIPLCS BooJc III,
the head, it appears that thefe (lars are generally of about the
fame apparent magnitude with the planets. But in regard that
their light may be often compared with the light of Satnm^
yea, and fome times exceeds it, it is evident that all comets in
their perihelions muft cither be placed below or not far above
Saturn ; and they are much miftaken who remove them al-
moft as far as the fixed ftars ; for if it was fo, the comets could
receive no more light from our fun than our planets do from
the fixed flars.
So far we have gone, without confidering the obfcmtition
which comets fuffer from that plenty of thick fmoke which
encompaffeth their heads, and through which the heads al-
ways ihew dull^ as through a cloud ; for by how much the
more a body is obfcured by this fmoke, by fo much the more
near it muft be allowed to come to the fun^ that it may vie
with the planets in the quantity of light which it reflet.
Whence it m probable that the comets defceod far below the
orb of Saturn, as we proved before from thdr parallax. But,
above al2^ the thing is evinced from their tails, which mnft he
owing either to the fun's light refledled by a fmoke aiifing
from them, and difperfing itfelf through the sther, or to the
light of their own heads. In the former cafe, we muft (horten
the diftance of the comets, left we be obliged to allow that
the fmoke arifing from their heads is propagated through
fuch a vaft extent of fpace, and with fuch a velocity and
expanfton, as will feem altogether incredible; in the latter
cafe, the whole light of both head and tail is to be afcribed
to the central nucleus. But, then, if we fuppofe all this light
to be united and condenfed within the difk of the nucleus,
certainly the nucleus will by far exceed Jupiter itfelf in fplen-
dor, efpecially when it emits a very large and lucid taU. If,
therefore, under a lefs apparent diameter, it reflefts more
light, it muft be much more illuminated by the fun, and
therefore much nearer to it ; and the fame argument will bring
down the heads of comets fometimes within the orb of Venus
viz. when, being hid under the fun's rays, they emit fuch huge
and fpendid tails, like beams of fire, as fometimes they do ; for
if all that light was fuppofed to be gathered together into one
Book IIL ' OF NATURAL PHILOSOPHY. 2,59
&Ar, it would fometimes exceed not one Venus only^ but a
great maoy fuch united into one.
Laftiy; the fame thing is inferred from the light of the
heads^ which increafes in the recefs of the comets from the
earth towards the fun^ and decreafes in their return from the
fun towards the earth ; for fo the comet of the year 1 665
(by the obfervations of Uevelius), from the time that it was
firfi feeo^ was always lofing of its apparent motion^ and there-*
fore had already pafTed its perigee; but yet the fplendor of
its head was daily increafing, till^ being hid under the fun's
rays, the comet ceafed to appear. The comet of the year
1683 (by the obfervations of the fame Hevelius)^ about the
end of Ju/yj when it firft appeared, moved at a very flow rate,
advancing only about 40 or 45 minutes in its orb in a day's
time ; but from that time its diurnal motion was continually
upon the increafe, till September 4, when it arofe to about 5
degrees ; and therefore, in all this interval of tim^ the comet
was approaching to the earth. Which is lik^ife proved
from the diaqieter of its head, meafured with a micrometer ;
ioiyAugufi 6, Hevtlim found it only 6' 05" ^ including the coma,
which, September 2, he obferved to be 9' 07*', and therefore
its head appeared far lefs abput the beginning than towards
the end of the motion ; though about the beginning, becaufe
nearer to the fun, it appeared fur more lucid than towards the
end, as the fame Hevelim deaares. Wherefore in all this
interval of time, on account of its recefs from the fun, it de-
oreafed in fplendor, notwithftanding its accefs towards the
earth. The comet of the year l6 1 8, about the 'midd/e of De-
cember, and that of the year 1680, about the end of the fame
month, did both move with their greateft velocity, and were
therefore then in their perigees ; but the greateft fplendor of
their heads was feen two weeks before, when they had juft
got clear of the fun's rays ; and the greateft fplendor of their
tails a little more early, when yet nearer to the fun* The
head of the former comet (according to the obfervations of
Cyfatus), December 1, appeared greater than the ftars of the
firft magnitude ; and, Decem6er]6 (then in the perigee), it was
but little dimioifhed in magnitudej but in th^ fplen4Qr and
S2
2G0 MATllEMATICAL PKINCIPLES Book HI.
brightnefs oF its light a great deal. January 7f Kepler, being
unceriaiii about tlie licad, left'off obferving. December \%
the head of the latter comet was feen and obferved by Mr.
Flamftedi when but 9 degrees diftant from the fun ; which
is fcarcely to be done in a ftar of the third magnitude. Dt'*
cembcr \5 and 17> it appeared as a flar of the third magni*
tude^ its luftre being diminiihed by the brightnefs <rf th6
clo^ near the fetting fun* December 26^ when it moved
with the greateft velocity^ being almoft in its perigee^ it was
lefs than the mouth of Pegafus, a ftar of the third magnitude.
January 3, it appeared as a ftar of the fourth. Jamuary Q, as
one of the fifth. January 13^ it was hid by the fplendoc of the
moon^ then in her iucreafe. January ^5, it was fcarcely
equal to the ftars of the fevenih magnitude. If we compare
equal inter\'als of time on one fide and on the other from the
perigee^ we ftiall find that the head of the comets which at
both intervals of time was far^ but yet equally^ temoved finom
the earth, tod fliould have therefore flione with equal fpled*
dor, appeared brighteft on the fide of the perigee towards' the
fun^ and difappeared on the other. Therefore, from the great
difference of light in the one fituation and in the other, we
conclude the great vicinity of the fun and comet in the for-*
mer ; for the light of comets ufes to be regular, and to appear
greateft when the heads move fafteft, and are therefore in
their perigees ; excepting in fo far as it is increafed by their
nearnefs to the fun.
Cor. 1. Therefore the comets ftiine by the fun's lights
which they reflect.
CoR. 2. From what has been faid, we may likewife under-
ftand why comets are fo frequently feen in that hemifpfaere
in which the fun is, and fo feldom in the other. If they were
vifible ill the regions far above Saturn, they would appear
more frequently in the parts oppofite to the fun ; for fuch as
were in thofe parts would be nearer to the earth, whereas the
prefence of the fun muft obfcure and hide thofe that appear
in the hemifpbere in which he is. Yet, looking over the hif-
'tory of comets, I find that four or five times more have been
feen in the hemiiphere towards the fun than in the oppofite
Book III. OF NATURAL PHILOSOPHY. 26l
hemifphere ; befides, without doubt^ not a few^ which have
been hid by the light of the fun : for comets defcendiog into
our parts neither emit tails, nor are fo well illuminated by the
fun, as to difcover th^mfelves to our naked eyes, until they
are come nearer to us than Jupiter. But the far greater part
of that fpherical fpace, which is defcribed about the fun with,
fo fjnall an interval^ lies on that fide of the earth which re-
gards the fun ; and the comets in that greater part are com-
monly more ftrongly illuminated, as being for the mo^ part
nearer to the fun.
Cor. 3. Hence alfo it is evident that the celeilial fpaces
are void of refinance;, for though the comets are carried ia
oblique paths, and fometimes contrary to the courfe of the
planets, yet thej' move every way with the greateft freedom,
and preferve their motions for, an exceeding long time, even
where contrary to the courfe of the planets. I am out in my
judgment if they are not a fort of planets revolving in orbits
returning into themfelves with a perpetual motion ; for, as to
.what fome writers contend, that they are no other than me-
teors, led into this opinion by the perpetual changes, that
.happen to their heads, it feems to have no foundation;, for
the heads of comets are encompaiTed with huge atmofpheres,
and the lowermoft parts of thefe atmofpheres muft be the
denfeft ; and therefore it is in the clouds only, not in the bo-r
dies of the comets themfelves, that thefe changes are fee^«
Thus the earth, if it was viewed from the planets, wo^W*
without all doubt, fhine by the light of its clouds, an^' the
Xolid body would fcarcely appear through the furr^unding
clouds. Thus alfo the belts of Jupiter are fornud in the
dpuds of that planet, for they change their po&^on one to
another, and the folid body of Jupiter is hardy to be feen
•through them ; and much more muft the bod>^s of comets be
hid under their atmofpheres, which are <^oth deeper ai^d
thicker.
PROPOSITION XL. THEOREM XX.
.That tlie comets move infomi of the coiic feSions, having their
foci in the centre of the fun ; and by radii. drawn to the fa^
defcribe areas proportional to the ti mes.
tGQ MATHEMATICAL PRINCIPLES Book lit.
This propofition appears from cor. I» prop. IS, book 1^ com-
pared with prop. 8^ 12^ and 13^ book 3.
Cor. 1 . Hence if comets are revolved in orbits returniDg
into themfelves, thofe orbits will be ellipfes ; and their perio*
die times be to the periodic times of the planets in the fefqni-
plicate proportion of their principal axes And therefore the
comets, which for the mod part of their courfe are higher
than the planets, and upon that account defcribe orbits with
greater axes, will require a longer time to finifli their revolu-
tions. Thus if the axis of a comet's orbit was fonr times
greater than the axis of the orbit of Saturn, the time of the
revolution of the comet would be to the time of the revolution
of Saturn, that is, to SO years, as 4 \/ 4 (or 8) to 1, and would
therefore be 240 vears
Cor. 2. But their orbits will be fo near to parabolas, that
parabolas may be ufed for them without feniiUe error.
Cor. 3. And, therefore, by cor. 7, prop. l6, book J, the
velocity of every comet will always be to the velocity of any
planet, fuppofed to be revolved at the fame diftance in a cir-
cle about the fun, nearly in the fubduplicate proportion of
double the diftance of the planet from the centre of the fun
to the diftance pf the comet from the fun's centre, very
ni^^arly. Let us fuppofe the radius of the orbis magnuSy or the
greateft femi-diameter of the ellip6s which the earth defcribes,
V) confift of 100000000 parts; and then the earth by its
rotan diurnal motion will defcribe 1720212 of thofe parts, and
7l6/5| by its horary motion. And therefore the comet, at
the fane mean diftance of the earth from the fun, with a
velocity Trhich is to the velocity of the earth as • 2 to 1,
would by its diurnal motion defcribe 2432747 parts, and
1013641 pai*3i by its horary motion. But at greater orlefs
diftances both \he diurnal and horary motion will be to this
diurnal and horaty motion in the reciprocal fubduplicate pro-
portion of the diftai^ces, and is therefore given.
Cor. 4. Wherefoie if the latus redtum of the parabola is
quadruple of the radium of the orbis magnuSy and the fquare
of that radius is fuppofej to confift of 100000000 parts, the
Book III. ; OF NATURAL PHILOSOPHY, ^63
. area which the comet will daily defcribe by a radius drawn
to the fan will be 12l6373i parts^r and the horary area will
be 50682J parts. But, if the latm reShm^ is greater or lefs
in any proportion, the diurnal and horary area will be lefs or
greater in the fuBduplicate of the fame proportion recipro-
cally.
LEMMA V.
Tojittd a curve line of the parabolic kind which Jhall pafi
^ through any given number of points. (PI, 15, Fig. 3.)
Let thofe points be A, B, C, D, E, F, &c, and from the fame
to any right line HN, given in pofition, let fall as many
perpendiculars AH, BI, CK, DL, EM, FN, &c.
Case 1.' If HI, IK, KL, &c. the intervals of the points
H, I, K, L, M, N, &c. are equal, take b, 2b, 3b, 4b, 5b, &c. the
firft differences of the perpendiculars AH, BI, CK, &c.;
their fecond differences c, 2c, Sc, 4c. fic/c. ; their third, d, 2d,
3d, &c. that is to fay, fo as AH — BI may be = b, BI — CK
= 2b, CK — DL = Sb, DL + EM = 4b,— EM + FN =
5b, &c. ; then b — 2b = c, &c. and fo on to the laft dif-
ference, which is here f. Then, ere6ling any perpendicular
RS, which may be confidered as an ordinate of the curve
required, in order to find the length of this ordinate, fuppofe
the intervals HI, IK, KL, LM, &c. to be units, and let AH
=: a, — HS = p, Ip into — IS = q, j-q into + SK = r, \i
into + SL = s, |s into + SM = t; proceeding, to wit, to
ME, the laft perpendicular but one, and prefixing negative
figns before the terms HS, IS, 8cc. which lie from S towards
A ; and affirmative figns before the terms SK, SL, &c. which
lie on the other fide of the point S; and, obferving well the
figns, RS will be =z a + bp + cq + dr + es + ft, + &c.
Case 2. But if HI,' IK, 8cc. the intervals of the points
H, I, K, L, &c, are unequal, take b, 2b, 3b, 4b, 5b, &c, the
firft differences of the perpendiculars AH, BI, CK, &c. divid-
ed by the intervals between thofe perpendiculars; c, 2c, Sc,
4c, &c. their fecond differences, divided by the intervals
between every two ; d, 2d, 3d, &c. their third differences,
divided by the intervals between every three ; e, 2e, &c. then:
fourth differences, divided by the interval between ^Very
84
294 MATIIEMATtCAX rBlNCIPLBS BookiU.
four; and fo forlli ; Xliat is, in fuch manner, that b may be =
All — BI „. BI — CK . CK — DL . .
_j_^_,2b = —j^j— , 3b = — gj— ., &c then c =
b — fib ^ 2b — 3b 3b— 4b- ,, . c— 2c,
_j_j^, 2c = -j^. 3c = -gjg-, Jkc. then d = -,g-..
^C — *)€
Cd = ' ^ ' , 8cc. And ihofe differeDces being founds let
AH be = a, — HS = p, p into — IS = q^ q into + SK =
r, T into + SL = s, s into + SM = t; proceeding; to wit,
to ME, the laft perpendicular but one ; and the ordinate RS
will be *= a + bp + cq + dr + es + ft, + &c.
Cor. Hence the areas of all curves may be nearly found;
for if fonie number of points of the curve to be fquared are
found, and a parabola be fuppofed to be drawn through thofe
points, the areaof this parabola will be nearly the fame with the
area of the curvilinear figure propofed to be fquared : but the
parabola can be always fquared geometrically by methods
vulgarly known.
LEMMA VI.
Certain obfervtd places of a comet being given, to find the
place of the fame to any intermediate given time.
Let HI, IK, KL, LM (in the preceding Fig.)> reprefent
the times between the obfervations ; HA, IB, KC, LD, ME,
five obferved longitudes of the comet; and HSthe given time
between the firfl obfervation and the longitude required.
Then if a regular curve ABCDE is fuppofed to be drawn
through the points A, B, C, D, E, and the ordinate RS is found
out by the preceding lemma, RS will be the longitude required.
After the fame method, from five obferved latitudes, we
may find the latitude to a given time.
If the differences of the obferved longitudes are iinall, fup-
pofe of 4 or 5 degrees, three or four obfervations will Be
fiifficient to find a new longitude and latitude ; but if the
differences are greater, as of 10 or 20 degrees, five obferva-
""tions ought to be ufed.
LEMMA VII.
Jlirough a g^ven point P (PL 15, Fig. 4) to draw a right Une
BC, whofe parts PB^ PC, cut off by two right lines AB, AC,
;4
■• -tf
} ,
«V"»
#
X^17 l^l.Jl: /utffc ^(S J.
liya
//ir. a
^5f- 3.
b 5?b ab -4> c/b
c 2c 3c 4c
d 2d 3d
e 2e
I \
^\C
M^.4
M N
Bdok Yti. dF NATURAL PHILOSOPHY-. 46S
given inpofitioHf may be one to the other in a given propor^
tion. . •
T'rom the given point P fappofe any right lit^ PD to te
drawn to either of the right lines given^ as A6 ; add products
the 'feme towards AC, the other given right line, as far as E^
fo as PE may be to PD in the given proportidn. Let EC be
pamllel to AD. Draw CPB, and PC will be to PB asPE
toPD. Q.KF,
LEMMA Vin.
Let ABC (PL 16, Fig. 1) be a parabola, having itsfoQus in S.
^1/ . the chdrd AC bifeded in I cut off' the fegment ABCl>
whofe diameter is Ip, anc^ vertex /x. In I/x produced take
fiO equal to one halfofliL* Join OS, and produce it to ^»
fo as S^ may be equal to 2S0. Now, fuppofing a comit io
revolve in the arc CBA, draw ^B, cutting AC in E ; I fay,
the, point E will cut off from the chord AC the fegment A^,
nearly proportional to the time.
For if we join £0, cutting the parabolic arc ABC in V, and
draw /xX touching the fame arc in the vertex /x, and meeting
EO in X, the curvilinear area AEX/xA will be to the curvili-
near area ACY/xA as AE to AC ; and, therefore, fihoe the
triangle ASE is to the triangle ASC in the fame proportion^
the whole area ASEX|xA will be to the whole area ASCY|»^
as AE to AC. But, becaufe ^O is to SO as 3 to 1, and fiO
to XO in the fame proportion, SX will be parallel to EB ;
and, therefore, j6inin]g BX, the triangle SEB will be equal td
the triangle XEB. Wherefore if to the area ASEXf»A'ire
add the triangle EXB, and from the fum fubdd6): the tnangie
SEB, there will remain the area ASBX^A, equal to the area
ASEX/mA, and therefore in proportion to the area ASCY^aA as
AE to AC. But the area ASBY^A is nearly equal to the Atea
ASBXftA ; and this area ASBY^iA is to the area ASCY^ A "as
thetimeof defcriptibn of the arc^AB to the time of deferip*
tion of the whole arc AC; and, therefore, AE is to AC
nearly in the proportion Of the' times. Q.E.D.
Cob. When the t>oint B falls u^on the vertex ft of the pa-
rabola, AE is to AC accurately in the proportion of the times.
fSO MATHEMATICAL PRINCII^LES Book IIL
SCHOLIUM.
If we join ft^ cutting AC in i, and in it take ^n in propor-
tion to fubB as 27MI to iSlMTft^ and draw Bn^ this Bn will cut
the cliord AC^ in the proportion of the times^ more accarateljr
than before; but the point n is to be taken beyond or on this
fide the point ^^ according as the point B is more or Ie& dif-
tant from the principal vertex of the parabola than tiie
point |x.
LEMMA IX.
AIC
The right lines Ifju and ftM, and the length ^ > are equal
among themfelvei.
For 4Sft is the latus re6lum of the parabola belonging to
the vertex ft.
LEMMA X.
Produce Sp* to 'Sand P (PI. l6, Fig. Q),fo as ilS may be one
third ofyj,, and SP may be to SN as SN to Sfi,; and in the
time that a comet would defcribe the arc AftC^ if it was
fuppofed to move alxoays forwards with t/ie velocity which
it hath in a height equal to S£^ it would defcribe a length
equal to the chord AC.
For if the comet with the velocity which it hath in p, was
in the faid time fuppofed to move uniformly forwards in the*
right line which touches the parabola in /x, the area which it
would defcribe by a radius drawn to the point S would be
equal to the parabolic area ACSfjuA ; and therefore the fpacc
Contained under the length defcribed in the tangent and the
length SpL would be to the fpace contained under the lengths
AC ahd SM as the area ASC/xA to the triangle ASC, that is,
as SN to SM. Wherefore AC is to the length defcribed in
the tangent as S^ to SN. But fince the velocity of the co-
met in the height SP (by cor. 6, prop. I6, book 1) is to the
. velocity of the fame in the height S/x in the reciprocal fub-
dupl jcate proportion of SP to Sjx, that is, in the proportbn of
Spt, to SN, the length defcribed with this velocity will be to
the length in the fame time defcribed in the tangent as Sft to
SN. Wherefore fince AC, and the length defcribed with
this new velocity, are in the fame proportion to the length
Book in. OF NATURAL PHILOSOPHY. '^87
defcribed ini the tangent^ they mnft be equal betwixt them-
felves, Q.E.D;
Cor. Therefore a cornet^ with that velocity which it hath
in the height S/x + |T|x> would in the fame time defcribe the
chord AC nearly.
LEMMA XI.
If a comet void of all motion was let fall from the height SN,
or Sii, + 4-T|x^ towards the fun, and was Jiill impelled to the
fun by the fame force uniformly continued by which it was
impelled atfrji, the fame, in one half of that time in which
it might defcribe the arc AC in its own orbit, would in &-
fcending defcribe afpace equal to the length /|x.
For in the fame time that the comet would require to de-
fcribe the parabolic arc AC, it would (by the laft lemma)^
with that velocity which it hath in the height SP, defcribe
the chord AC; and, therefore (by cor. 7, prop. l6, book 1), if
it was in the fame time fuppofed to revolve by the force of its
own gravity in a circle whofe femi-diameter was SP, it would
defcribe an arc of that circle, the length of which would be
to the chord of the parabolic arc AC in the fubduplicate pro-
portion of 1 to 2. Wherefore if with that weight, which in
the height SP it hath towards the fun, it fhould fall from that
. height towards the fun, it would (by cor.- 9, prop. 4, book 1)
in half the faid time defcribe a fpace equal to the fquare of
half the faid chord appHed to quadruple the height SP, that
AI*
is, it would defcribe the fpace Top* ^^^ ^^^^^ ^^ weight of
the comet towards the fun in the height SN is to the weight
of die fame towards the fun in the height SP as SP to Spi, the
con^et, by the weight which it hath in the height SN, in fidl*
ing from that height towards the fun, would in the iame time
AI*
defcribe the fpace -^ * that is, a fpace equal to the length
Iftor ixM. Q.E.D.
PROPOSITION XLI. PROBLEM XXL
From three obfervatiom given to determine the orbit of a eouut
moving in a parabola. .
f06 UATBSM ATICAL rRIirCIPLBS Bttok HI.
T^is being a problem of very great difficulty^ I tried manj^
methods of refoiving it; and feveral of thofe problems^ the
eompofition whereof I have given in the firft book^ tended to
ibis purpofe. But aflerwards I qontrived the following folu-
tion^ which is i'omething more fimple.
Sele<5l three obiervations diibmt one from another by inter-
Yab of time nearly equal ; but let that interval of time in wbieb
ifae comet moves more llowly be fomewhat greater than the
other; fo^ to wit^ that the diffcreiice of the times may be to
4be fum of the times as the fum of tlie times to about 60O
days; or that tlie point E (PI. l6> Fig. 1) may fall upon M
nearly, and may err therefrom rather towards I than towards
A. If fuch dire^l obfervations are not at hand^ a new place
oi the comet mufl; be found, bv lem. 6.
Lets (PI. 16, Fig. 3) reprefeiU the fun; T, t, t, three
plaeesof the earth in the orbismagnus; TA, ffi^ rC, three
^bfenred longitudes of the comet; V the time between the
tiirft obfervation and the fecond ; W the time between the
'feoond and the third; X the length which ip the 'whole time
-V + W the comet might defcribe with that velocity which
it hath in the mean diftance of the earth from the fun^ which
length is to be found by cor. 3, prop. 40, book 3 ; and tV a
perpendicular upon the chord Tt. In the mean obferved
kngitude tB take at pleafure the point B, for the place
of the comet in the plane of the ecliptic ; and from thence,
towards the fun S, draw the line BE, which may be to the
perpendicular tV as the content under SB and St* to the cube
lof the hypothenufe of the right angled triangle, whofe fide*
are SB, and the tangent of the latitude of the comet in the
'fecond obfervation to the radius tB. And through the point
E (by lemma 7) draw the right line AEC, whofe parts AE
and EC, terminating in the right lines TA and tC, may be
one fo the other as the times V and W: then A and C wril
be nearly the places of the comet in the plane of the ecliptic in
the firft and thii'd obfervations, if B was its place rightly
^ailumedin the fecond.
Upon AC, bifedled in I, ereft the perpendicular li. Through
B draw the obfcure line Bi parallel to AC Join the obfcure
Book IIL OF NATURAL PHILOSOPHY* 2^9
line Si, cutting AC in A, and complete tbe parallelo^am il hpu^
Take !«• equal to SU; and through the fun S dsaw the obfcure
line 0-^ equal to SS^ -f 3 iA. Then> cancelling the letter
A, E, C, I, from the point B towards the point J, draw the
new obfcui^ line 6£^ which may be to the former BE in the
duplicate proportion of the diftance BS to the quantity S^ 4
i iA. And through the point E draw again the right line
AEC by the fame rule as before^ that is^ fo as its parts A£ and
£C may be one to the other as the times V and W betiw^ea
the obfervations. Thus A and C will be the places of the
comet more accurately.
Upon AC, bifefted in I^ereiSl the perpendiculars AM, CN,*
lO, of which AM an'^ CN may be the tangents of the latitudes
in the firft and third obfervations, to the radii TA and rC.
Join MN, cutting lO in O. Draw the reftangular parallelo-
gram ilAffc, as before. In IA produced take ID equal to Sft
+ 4 iA. Then in MN, towards N, take MP, which may h6
to the above found length X in the fubduplicate proportion of
the mean diftance of the earth from the fun (or of the femi*
diameter of the orbis magnvs} to the diftance OD. If the
point P fall upon the point N; A, B, and C, will be thitje
places of the comet^ through which its orbit is to be defcribed
in the plane of the ecliptic. But if the point P falls not upon
the point N, in the right line AC take CG equal to NP, fo
as the points 6 and P may lie on the fame fide of tbe
lineNC.
By the fame method as the points E, A, C, 6, were found
firom the afiumed point B, from other points b and /) aflumili
at pleafure, find out the new points e, a, c, g ; and 5, a, k, y.
Then through G, g, and y, draw the circumference of a cirdjt
Ggy, cutting the right line tC in Z : and Z will be one plaoe
of the comet in the plane of the ecliptic. And in AC, ac, ok^
taking AF, af, «(p, equal refpeftively to CG, eg, jty ; throndh
the points F, f, and (p, draw the circumference of a etrfib
Ff(p, cutting the right line AT in X ; and the point X will ))e
another place of the comet in the plane of the ecliptic. And
at the points X and 7a, ere^ing the tangents of the laiilades
ef the comet to the radii TX and tZ^ two places fX iStm
170 MATHEMATICAL PBINCIPLX8 Book IIL
comet in its own orbit will be determiDed. Laftly^ if (by
prop. \9, book I) to the focus S a parabola is defcribed paff-
iog throngh thofe two places^ this parabola will be the orbit
of the comet. Q.E.I.
The demonftration of this conftru6lioD follows from the
preceding lemmas^ becaufe the right line AC is cut in E in
the proportion of the times^ by lem. 7, as it ought to be^ by
lem. 8 ; and BE^ by. lem. 11^ is a portion of the right line BS
or B| in the plane of the ecliptic^ intercepted between the
ire ABC a»d the chord AEC; and MP (by cor. lem. 10) is
ihe length of the chord of tliat arc^ which the comet (hould
4efcribe in its proper orbit between the iirft and third ob-
fervation, and therefore is equal to MN> providing B is a true
place of the comet in the plane of the ecliptic.
But it will be convenient to aflfume the points B^ b^ fi, not
at random^ but nearly true. If the angle AQt, at which the
proje£Uon of the orbit in the plane of the ecliptic cuts the
right line tB^ is rudely known^ at that angle with Bt draw the
obfcure line AC^ which may be to yXr in the fubduplicatQ
proportion of SQ to St ; and^ drawing the right line SEB fo
a|^ its part £B may be equal to the length Vt^ the point B
will be determined, which we are to ul'e for the firft time.
Then, cancelling the right line AC, and drawing anew AC
according to the preceding conftruftion, and, moreover, find-
ing the length MP, in tB take the point b, by this rule, that,
if TA and tC interfe6l each other in Y, the diftance Yb may
be to the diftance YB in a proportion compounded of the
l^portion of MP to MN, and the fubduplicate proportion of
SB to Sb. And by the fame method you may find the third
point /5, if you pleafe to repeat the operation the third time ;
hot if this method is followed, two operations generally will
be fufficient ; for if the diftance Bb happens to be very fmall,
after the points F, f, and G, g, are found, draw the right lines
FJPand Gg, and they will cut TA and tC in the points requir-
ed, X and Z.
EXAMPLE.
Let the comet of the year 1680 be propofed. The follow-
ing table ftiews the motion thereof, as obferved by Flamjled,
Ti'
FlatfXVI. FoLJ^. ,^uz^^^ .
^>.3
--■•
«^
#
.4
\
it
f
4
I
It
^
m
Book in. OF NATUltAL PHILOSOPHY. 2?!
and calculated afterwards by him from his obfervatlons^ and
corre6led by Dr. Halley from the fame obfervations.
Time
Appar.
l680,Dec. 12
21
24
26
29
30
5
/»
iSSl^Jion.
h.
4.46
6.32|
6.12
5.14
7.55
8.02
5.51
10
25
96.49
5.54
13 6/56
7.44
leb.
30 8.07
216.20
5"6.50
True?
4.46.
6,36.59
6.17.52
5.20.44
8.03.02
8.10.26
6.01.38
7.00.53
6.06.10
7.08.55
7.58.42
8.21.53
6.34.51
7.04.41
Stui's
Longitude
n
Vf 1.51.23
11.06.44
14.09.26
16.09.22
19.19-43
20.21.09
26.22.18
::: 0.29.02
1.27.43
4.33.20
16.45.36
21.49.58
24.46.59
27.49.51
Comet's
Longitude.
^
ti
K
r
2028
5326
6.32.30
5.08.12
1 8.49.23
28.24.13
13.10.41
17.38.
8.48.5
18.44.04
20.40.50
25«59.48
9.35.
13.19.51
15.13.53
\6.59'(^
Lat. N?
8.28.
21.42.13
25.23. 5
27.00.52
28.09.58
.11.53
.15. 7
24.11.56
23.43.52
22.17.28
17.56.30
16.42.18
16.04. 1
15.27. 3
To thefe you may add fome obfervations of mine.
Ap.
iTim«.
l681,fe^. 25
27
Mar, 1
2
h. ,
8.30
8.15
11.
8.
5111.30
9.30
8.30
7
9
Comet's
Longitude.
II
.« 26. 18.35
27.04.20
27.52.42
28.12.48
29. 18.
n 0. 4.
0.43. 4
Lat. N.
J 2.46.46
22.36.12
12.23.40
12.19.38
12.03.16
11.57.
11.45.52
Thefe obfervations were made by atelefcope of 7 feet, with
a micrometer and threads placed in the focus of the telje-
fcope ; by which inflruments we determined the pofitiolis
both of the fixed flars among themfelves, and of the comet
in refpedl of the fixed ftars. Let A (PI. 17) reprefeut tjte
ftar of the fourth magnitude in the left heel of Perfeui
(Bayer*s o), B the following ftar of the third magnitude in
th^..left foot (Bajfer's^, C a ftar of the fixth magnito^
(Bayer's n) in the heel of the fame foot, and D, E, F, G, H,
J, K, L, M, N, O, Z, a, fi, y, i, other iinaller ftars in t]ie
fame foot ; and let p, P, Q, R, S, T, V, X, reprefent the
places of the comet in the obfervations above fet down ; and^
9K MATIIBMATICAL MtWClThtB Sook W^
leckoDiog tbe diftaace AB of 80^ parts^ AC was 521 of thofq
parts; BC, 38^; AD, 57A; BD, 8^; CD, 23f ; AE, 89$.;
C£, 57 i; DB,49«; Al, 27i?r; »•> 52i; CI, 36^; J>h
5SA; AK,38|; BK,43; CK,3I|; FK,29; FB,M; PC,
36i; AH, 18|; DH, 50f; BN,46A; CN, 31|; BL, 45^;
NL> 317. HO waj» io HI as 7 to 6, aod^ produced, did pafs
between the ftars D and E, fo as the diftance of the ftar D
from this right line was |>CDl LM was to LN as 2 to 9, and,
piHKluced, did pafs through the ftar H. Thus were the pofi-
tions of the fixed ftars determined in refpe<% of one another.
Mr. Pound has (ince obferved a fecond time the pofitions
of thefe fixed ftars amongft themfelves, and colIe6led their
longitudes and latitudes according to the following table.
The
fixed
(lars.
Their
Longitudes.
Latitude
North.
The
fixed
ftars.
Their
Longitudes.
Latitude
North.
A
B
C
E
F
G
H
I
K
a 26.41.50
28.40.23
27.58.30
26.27.17
28.28.37
26.56. 8
27.11.45
27.25. 2
-5^27.42. 7
1 II
12. 8.36
11.17.54
12.40.25
12.52. 7
11.52.22
12. 4.58
12. 2. 1
11.53.11
11.53.26
L
M
N
Z
Y
i
« 29.33.34
2.9.1vS.54
28.48.25)
29.44..48
29.52. 3
n 0. 8.23
0.40.10
1. 3.20
/ It
12. 7.46
12. 7.20
12^1. 9
11.57.13
11.55.48
11^8.56
11.55.18
11.30.42
The pofitions of the comet to thcfc fixed liars were obferv-
ed to be as follow :
Frida)'^, February 25, O.S. at 8p. P. M. the diftance of the
comet in p from the ftar E was lefs than ^AE, and greater
than iAE, and therefore nearly equal to tVAE ; and the an-
gle ApE was a little obtufe, but almoft right. For from A,
letting fall a perpendicular on pE, the diftance of the comet
from that perpendicular was l^pE.
The fame night, at 9|^. the diftance of the comet in P
firom the ftar E was greater tlran — AE, and lefs than -7
AE, and therefore nearly equal to — of AE, or ^AE. But
. '-Ay
k
^
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Jiook III; OF NATUItAL PHILOSOPHY. £75
the diftiokoe of tbe comet from the perpendicular let fall from
the ftar A upon the right line P£ was ^PE.
Sunday^ February 27, 8i^P. M* the diftanceof the comet
in Q from the ftar O was equal to the diftance of the flars O
had H ; and the right line QO produced pafled between the
fiars K and B. I could not^ by reafon of intervening clouds^
determine the pofition of the ftar to greater accuracy.
Tuefday^ March 1, 1 J^. P. M. the comet in R lay exa6Uy
in a line between the ftars K and C^ fo as the part CR of the
jjght line CRK was a little greater thw fCK^ and a little led
than 4€K + fCR, and therefore =fCK+VrCR, or J|.CK.
Wednefday, March % 8^. P. M. the diftance of the comet
jn S from the ft;ar C was nearly ^FC ; the diftance of the ftar
P from the right line CS produced was ^FC ; and the dif*
tance of the ftar B from the fame right line was five timef
greater than the diftance of the ftar F ; and the right line
NS produced paffed between the ftm^ H and I five or fix
times nearer to the ftar H than to the ftar I.
Saturday^ March 5, llf**. P. M. when the comet was in T,
the right line MT was equal to {ML^ and the right line LT
iNCpduoed paflei between B and F four or five times nearer to
W than to B^ cutting off from BF a fifth or fixth part thereof
Awards F : and MT produced paffed on the outfide of the,
IfMce BF towards the ftar B four times nearer to the ftar B
llMin to the ftar F. M was a very finall ftar^ fcarcely to be
leen by the tekfcope ; but the ftar L was greater^ and of about
4he eighth magnitude.
Monday^ March 7, 9?» P. M. the comet being in V, tlie
right line Va produced did pafs between B and F^ cutting.off^
' ^fom BF towards F^ ^ of BF^ and was to the right line V$
■ M 5 to 4. And the diftance of the comet from the right line
. ^ was 4V/5.
i- Wednefday, March 9, S-f*. P. M. the comet being in X, .
ilut right line 7X was equal to jyS ; and the' perpendicular
I0I ikU from. the ftar t upon the right 7X was f of yl>
1 Xhe lame nighty at 12K the comet being in Y^ the right
Jine 7Y was equal to j- of yS^ or a little lefs^ as perhaps -^ of
,g4; and a perpendicular letfaUj&om the jftar I pn the right
Vol. II. T
£74 MATHEMATICAL PEINCIPLES Book HI.
]ine 7Y was equal to about | or f yi. But the comet being
then exlrcmely near the horizon^ was fcarcely difcemible^ and
therefore its place could not be determined with that certainty
as in the foregoing obfervations.
From thefe obfervations, by conftrudlions of figures and
calculations, I deduced the longitudes and latitudes of the
comet ; and Mr. Pound, by correding the places of the fixed
ftars, hath determined more corredlly the places of the comet,
which corre<5l places are fet down above. Though my micro-
meter was none of the heft, yet the errors in longitude and la-
titude (as derived from my obfervations) fcarcely exceed one
minute. The comet (according to my obfervations), about the
end of its motion, began to decline fenfibly towards the north,
from the parallel which it defcribed about the end of f!r-
bruary.
Now, in order to determine the orbit of the comet out of
the obfervations above defcribed, I feleded thoie three which
Flam/led made, Dec. 21, Jan. 5, and Jan. 25 ; from which
I found St of 9842,1 parts, and Vt of 455, fuch as the femi-
diameter of the orbis magnus contains 10000. Then for the
firft obfervation, affuming tB of 5657 of thofc parts, I found
SB 9747, BE for the firft time 412, S/x 9503, iX 413, BE for
the fecond time 421, OD 1 0186, X 8528,4, PM8450, MN8475,
NP 25 ; from whence, by the fecond operation, I coUe^ed
the diftance tb 5640; and by this operation I at laft deduced
the diftances TX 4775 andrZ 1 1322. From which, limiting
the orbit, I found its defcending node in ss, and afcending
node in >f 1° 53' ; the inclination of its plane to the plane of
the ecliptic 6I® 20J.' ; the vertex thereof (or the perihelion of
the comet) diftant from the node 8° 38', and in t 27® 43*;
with latitude 7** 34' fouth ; its latus re&um 236,8 ; and the di-
urnal area defcribed by a radius drawn to the fun 93585, fup?
pofing the fquare of the femi-diameter of the orbis magnus
100000000; that the comet in this orbit moved dire6Uy ac-
cording to the order of the figns, and on Dec. 8*. Otf*. 04
P. M. was in the vertex or perihelion of its orbit* All which
I determined by fcale and compafs, and the chords of anglesii
taken from the table of natural fines, iu a pretty large figure
Book III. OP NATUBAL PHILOSOPHY. 275
in whicb^ to wit^ the radius of the orbis magnus (confifling of
ICXXX) parks) was equal to 16|- inches of an Englijk foot.
Laftly, in order to difcover whether the comet did truly
move in the orbit fo determined^ I inveftigated its places in.
this orbit partly by arithmetical operations, and partly by
fcale apd compafs, to the times of fome of the obfervations^
as may be feen in the following table.
r
Dec. 12
Febr. 5
Mar, 5
Dift.
from
fun.
Longitude
computed.
The Comet's
Latitude
comput-
ed.
.2792 Vf 6*^.32'
8403 X 13 '^H
16669|« 17.00
217371 29 .I9i
8M8|
28. 00„
15. 291
12. 4
Longitude
obferved. .
Vf 6°.31|
Xl3. Hi
Q9* 2o|.
Latitude
obierved*
8^26
28. 10^
15. 27|.
12. 34
Dif
Lo.
+ 1
+ 2
+
— 1
Dif.^
Lat.
-i(hV
+ 2i
But afterwards Dr. Halley did determine^ the orbit to a
greater accuracy by an arithmetical calculus than could he
done by linear defcriptions ; and^ retaining the place of the
nodes in ss and yf V* 5S', and the inclination of the plane
of the orbit to the ecliptic 6I® 9M/, as well as the time of the
comet's being in perihelio^ Dec. 8**. 00'*'. .04', he found the
diftance of the perihelion from the afcending node meafured
in the comet's orbit g^ 9Xi, and the latm reQum of the para*
bok £430 parts, fuppoiing the mean diftance of the fun from
the earth to be 100000 parts : and from thefe data, by an ac*
curate arithmetical calculus^ he computed the places of the
comet to the times of the pbfervations as follows.
■ .' » .•
•i
) .*: . •
Tc
MATHEMATICAL FltlfClPLBS Book III.
^i
a
^ — COOOOOOO-^OO'
14- I ++ I ++4.+ + I++ + .
■0»" — F«»- — oooo~c*wo«o
2S2SS
This comet alfo appeared in the November beforej and at
Coburg, in Saxony, was obferved by Mr. Gottfried Kirch,
on the 4th of that month, on the 6th and 1 1 th O^ S ; from
its pofitioQB to the neareft fixed fiars obferved with fufficient
accuracy, fomeUmes with a two feet, and fometimes with a
ten feet telefcope ; from the difference of longitudes of Co-
hurg and London, 1 1° ; and from the places of the fixed flars
obferved by Mr. Pound, Dr. Halley has determined the places
•f the comet as follows.
Book in. OS NATURAL PHILOSOPHY^ £77
Nov. 5f 17^- ^f apparent time at London, the cpmet was
in a 89 deg. 51'^ widi 1 deg. 17' 45" latitude norths
November 5, ISf" 58' the comet was in ifR 3^ 23', with 1^ S
north lat.
Hovcmbtr 10, l€^. 31', the comet was equally diftant from
two ftars in $t> which are 9 and r in Bayer ; but it had not
quite touched the right line that joins them, but was very
little diftant from it. In Flam/ied^s catalogue this ftar 9 was
then in 1% 14°. 15', with 1 deg. 41' lat. north nearly, and t
in itR 17*^ 3¥ with deg. 34' lat. fouth ;^.and the middle point
between thofe ftars was m 15° 39i', with 0° 334' lat. north.
Let the diftance of the comet from that right line be about
10' or 12* ; and the difference of the longitude of the comet
and that middle point will be 7' ; and the difference of the la-
titude nearly 71' ; and thence it follows that the comet was
in m 15° 32', with about 26' lat. north.
The &t& obfervation from the pofition of the comet with
refpe6l to certain fmall fixed ftars had all the exadnefs that
could be defired ; the fecond alfo was accurate enough. In
the third obfervation, which was the leaft accurate, there
might be an error of 6 or 7 minutes, but hardly greater. The
longitude of the comet, as found in the firft and moft accurate
obfervation, being computed in the aforefaid parabolic orbit,
comes out a 29"* 30' 22", its latitude north 1° 25' 7", and ite
diftance from the fun 115546.
Moreover, Dr. Halley, obferving that a remarkable comet
had appeared four times at equal intervals of 575 years (that
is, in the month of September after Julius Cafar was killed ;
An, Chr,53], in the confulate of Lampadius and Orejies ;
An. Chr. 1 106, in the month of February ; and at the end of
the year I68O ; and that with a long and remarkable tail, ex-
cept when it was feen after Cafqr^s death, at which time, by
reafon of the inconvenient fituation of the earth, the tail was
not fo confpicuous), fet himfelf to find out an elliptic orbit
whofe greater axis ihouldbe 1382957 parts, the mean diftance
of the earth from the fun containing 10000 fuch ; in which
orbit a comet might revolve in 675 years ; and, placing the
afcending node in ^ S^ 2', the inclination of the pla^e of the
T 3
278 MATHEMATICAL PRINCIPLES Book III.
orbit to the plane of the ecliptic in on angle of 6l^ S' 48"^ the
perihelion of the comet in this plane in f. 22^ 44' 25", the
equal time of the perihelion December 7^- 23*^. 9', tbediftance
of the perihelion from the afcending node in the plane of the
ecliptic 9° 17' 35'^ dnd its conjugate axis 18481>2, he com-
puted the motions of the comet in this elliptic orbit. The
places of the comet, as deduced from the obfervations, and as
arifing from computation made in this orbit^ may be feen in
the following table.
- c
• • • • • o •
-* O C9 H* 00 ^ OiOt
S X
H* I-* K> H*
B *^ 09 00 00 Ot O)
o
• •••••
C^ »0 K9 ^d )0 >t ^
O^ 0)ifi^ h^ )0 09
to
O OD 0)0 Ot 03
09 Q !-• oa 1^ «•
O tt*" O 00 -<I^
ii<¥
^ 09 H« fO 1^ 09
10 00 O llki ^ 00 )0
\f% i^ »-• fO i-> 09
*9 O *-• 09 09 KB O
O K) )0 »o »o fd
3^00 00 "^ 0» w 00
ft >-*^ O 09 to 00
Ot o« o« H*
^ 00 00 to Ot 00 O
i X II eg 3
)3^
Ot 09^ ^
09 ^^ ot «
to 09 I-*
OOP
O i-< w o
to H-
ooo*
l> MP
w w to
» ^ 09 00
00 0« C^09
to H^ i-i to o
00 00 00 o« 00^
>. 09 ^ to
O 00 H* h-
^ to H* »^
- ^i »^ to
o to to to
71 00 00 "^
»*t 09 to
^ Oi*- to
w Ot »— 09 to C;» *
O to 0^09 1^ »-•
09 i-< to »^
O •t^' O to
09 »— »^ 09 to ^ i
oi ot c/» to to to ^
to to
trt H* 00 to
•-- I-- O O •— H* O
a. !-• O to
K> t^ to to
09 *^t^iO
o» to o» to i-»
09 0^09 Ot O^^ '
;« 00 09
^ ^ 00 W
09 1^
ot to ClO
09 0« 00 ^
Ox <*> ^^^ to "
09 2
f
+ +I I I I
o o to
• • •
OO 09
^ 09 ►-
P o o >-
• • •
5 OiOVD
0« Ot 1-^
09 00 O
I +++++
O to ^
• • •
09 to
++ +
1-^ h- o -
• • •
00 to «
to to to "
g
B
k
as
p
S
c
I
8
l + l
•- o o
• • •
09^ 00
I.
a*
lO
a>
Bookllh OF KATUEAL PHILOSOPHY. 279
The obfervations of this comet from the beginning to the
end agree as perfedUy with the motion of the comet in the orbit
juft now defcribed as the motions of the planets do with the
theories from whence they are calculated ; and by this agree-
ment plainly evince that it was one and the fame comet that
appeared all that time^ and alfo that the orbit of that comet is
here rightly defined.
In the foregoing table we have omitted the obfervations of
Nov. 16, 18, 20, and 23,' as not fufiiciently accurate, for at
thofe times feveral perfons had obferved the comet. Nov. 17,
O. S. Ponthaus and his companions, at 6^. in the morning at
Rome (that is, 5^. 10' at London), by threads dire&ed to the
fixed ftars, obferved the comet in 1^^ 8® 30', with latitude 0'
40' fouth. Their obfervations may be feen in a treatife which
Pontfueus publiflied concerning this comet. Cellius, who
was prefent, and communicated his obfervaticHis in a letter
to Cajfini, faw the comet at the fame hour in ^ 8^ 30', with
latitude 0° 30' fouth. It was likewife feen by Galletius at
the fame hour at Avignon (that is, at 5^ . 42' morning at
London) in ^ 8° without latitude. But by the theory the
comet was at that time in d& 8° l6' 45", and its latitude was
0° 53'7" fouth.
Nov. 18, at 6^. 30' in the morning at Rome (that is, at 5**.
40' at London), PontJueus obferved the comet in ^ 13° 30',
with latitude l"* 20' fouth ; and Cellim in & 12!" 30', with
latitude 1° 00' fouth. But at 5^. 30' in the morning at Avig^
non, Galletius faw it in !£i 13° 00', with latitude 1° 00' fouth.
In the Univerfity of La Fleche, in France, at 5^. in the morn-
ing (that is, at 5^ 9' at London), it was feen by P. Ango, in the
middle between two fmall ftars, one of which is the middle of the
three which lie in a right line in the fou them hand ofVirgo, Bayers
4^; and the other is the outmoft of the wing, Bayer's^. Whence
the comet was then in a 12° 46', with latitude 50^ fouth. And
I was informed by Dr. Halley, that on the fame day at
Bofton in New England, in the latitude of 42| deg. at 5^. in
the morning (that is, at 9^. 44' in the morning at London),
the comet was feen near ^ I4°f with latitude 1^ SO" fouth*
T4
280 MATHfiM ATICAL PBINCIPLES BoiJk YLL.
Nov. ig, at 41^. at Cambridge, the comet (by the obfenratioQ
of a young man) was diftant from Spica «|l about ^ towatds
the north weft. Now the fpike was at that time in a 19^ 23/
AtT, with latitude <£" \ b^' fouth. The fame day^ at 5^. in the
mornings at Bofion in N^v England, the comet was diftant
irom Spica nx 1^ with the difference of 40' in latitude. The
fame day, in the ifland of Jamaica, it was about 1** diftant
firom Spica *R. The fame day, Mr. Arthur Starer, at the
river Patuxent, near Hunting Creek, in Maryland, in the con-
fines of Firginia, in lat 38i% at 5 in the morning (that is^ at
10^. at London), faw the comet above Spica *R^and very
nearly joined with it, the diftance between them being about
\ of one deg. And from thefe obfervations compared^ I
conclude, that at Q*^ 44' at London the comet was in ^ 18*
iXl, with about 1^ 0,5' latitude fouth. Now by the theory the
comet was at that time in €h ^W* 52! 15"^ with 1^ 2ff 64" Idt
fonth.
Nov. 20, Montenari. profeflbr of aftronomy at Padua, at 6^.
in the morning at Fenice (that is, 5^. Kf at London), faw the
comet in ^ 23% with latitude 1^ 30^ fouth. The fame day, at
Bofton, it was diftant from Spica r^ by about 4^ of longitude
eaft, and therefore was in ^ 23° 24' nearly.
"Nov. 21, Ponthatis and his companions, at 7i^« in the morn*
ing, obferved the comet in ^ 27° 50', with latitude 1^ I6'
fouth ; Cellius, in ^ 28°; P. Ango at 5^. in the morning, in ^
27^45'; Montenari in £ik 27° 51'. The fame day, in the ifland
o! Jamaica, it was feen near the beginning of nt, and of abonr
the fame latitude with Spica nji^ that is, 2° 2'. The fame day,
at 5^. morning, at Ballafore, in the Eaft Indies (that is, at ll**.
20' of the night preceding at London), the diftance of the
comet from Spica WR was taken 7° 35' to the eaft. It was in
a right line between the fpike and the balance, and there-*
fore was then in sCt 26° 58', with about 1*^ 11' lat. fouth ; and
after 5**. 40' (that is, at 5* .morning at London), it was in d
28° 12', with 1° 16' lat. fouth. Now by the theory the comet
was then in a 28° 10' 36", with 1° 5S' 35" lat. fouth.
Nov. 22, the comet was feen by Montenari in nt 2° S3' ;
but at Bojion in New England, it was found in about ni 3%
and with almoft the fame latitude as before, that is^ 1° 30'.
Book IlL OF NATUKAL PHILOSOPHY. Qgy
The fame day^ at 5^. morning at Ballafore, the comet was
obferved in ii| 1** 50'; and therefore, at 5*». morning at
London, the comet was in itt 3^ 5' nearly. The fame day, at
6|^. in the morning at London, Dr. Hook obferved it in a^ut
tti 3^ SCf, and that in the right line which paifeth through t^pictf
tfi and Cor Leonis; not, indeed, exadUy, but deviating a
little from that line towards the north. MonUnari likewife
obferved, that this day, and fome days after, a right line
drawn from the comet through Spica paifed by the fouth
fide of Cor Leonis at a very fmall diftance therefrom. The
right line through Cor Leonis and Spica rrji did cut the eclip*
tic in nfi S^ 4& at an angle of 2® 51' ; and if the comet had
been in this line and in tii 3"*, its latitude would have been
2^ 2& ; but fince Hook and Montenari agree that the comet
was at fome fmall diftance from this line towards the north,
its latitude mufi have been fomething lefs. On the 20th, bj
the obfervation of Montenari, its latitude was almoft the fame
with that of Spica, that is, about 1^ 30'. But by the agree-
ment of Hook, Montenari, and Ango, the latitude was con*
tinnally increafing, and therefore muft now, on the 2£d, be
fenfibly greater than 1^ 30^ ; and, taking a mean between the
extreme limits but now ftated, s"" £6' and 1^ 30', the latitude
will be about 1^ 58'. Hook and Montenari agree that the
tail of the comet was dire6^ed towards Spica vji,, declining a
little from that ilar towards the fouth according to Hook, but
towards the north according to Montenari ; and, therefore,
that declination was fcarcely fenfible ; and the tail, lying
nearly parallel to the equator, deviated a little from the op-
pofition of the fun towards the north.
Nov. 23, O. S. at 5* morning, at Nuremberg (that is, at
4? at London), Mr. Zimmerman faw the comet in Hi 8° 8^,
with 2^ 31' fouth iat. its place being colleAcd by taking its
diftances from fixed fi:ars.
Nov. 24, before fnn-rifing, the comet was feen by Monte"
nari in ni 12^ 52' on the north fide of the right line through
Cor Leonii and Spica i%, and therefore its latitude was fome-
thing lels than 2^ 38' ; and fince the latitude, as we faid, by
the concurring obfervations of Montenari, Ango, and Hook,
282 MATHEMATICAL PRINCIPLES Book IIT.
was continually increafing^ therefore^ it was now, on the 24th^
fomething greater than 1* 58' ; and> taking the mean qaan*
tity, may be reckoned 2** 18', without any confiderable error.
Pontlucus and Galletius will have it that the latitude was now
decreafing ; and Cellius, and the obfervcr in New England,
that it continued the fame, viz. of about 1^ or 1 j^ libe ob-
fervations of Ponthaui and Cellius are more rude, efpecially
thofe which were made by taking the azimuths and altitudes ;
as are alfo the obfcrvations of Galletius. Thofe are better
which were made by taking the pofition of the comet to the
fixed ilars by Monteuari, Hook, jingo, and the obferver in
New England, and fomctimes by Ponthaus and Cellius. The
fame day, at 5^ morning, at Ballafore, the comet was ob*^
ferved in ni 11** 46'; and, therefore, at 5**. morning at London,
was in nt 13° nearly. And, by the theory, the comet was at
that time in ni 13** 22' 42".
Nov. 25, before fun-rife, Mon^f//ari obferved the comet in
nt 17i^ nearly; and Cellius obferved at the fame time that
the comet was in a right line between the bright ftar in the
right thigh of Virgo and the fouthem fcale of libra; and this
right line cuts the comet's way in v\ 18** 36'. And, by the
theory, the comet was in nt iS^® nearly.
From all this it is plain that thefe obfcrvations agree with
the theory, fo far as they agree with one another; and by this
agreement it is made clear that it was one and the fame comet
that appeared all the time from Nov. 4 to Mar. Q. The path
of this comet did twice cut the plane of the ecliptic, and
therefore was not a right line. It did cut the ecliptic not in
oppofite parts of the heavens, but in the end of Virgo and be-
ginning of Capricorn, including an arc of about 98^ .; and
therefore the way of the comet did very much deviate from
the path of a great circle; for in the month of Nov. it de-
clined at leaft 3° from the ecliptic towards the fouth ; and in
the month of Dec. following it declined 29^ from the ecliptic
towards the north ; the two parts of the orbit in which the
comet defcended towards the fun, and afcended again from
the fun, declining one from the other by an apparent angle
of above 30*, as obferved by MontenaH. This comet travelled
Book UL OF NATURAL PHlLOSOPHr. 283
over 9 figns> to wit, from the lafi deg. of ^ to the beginning
of n^ befide the fign of A,^ through which it paiTed before
it began to be feen; and there is no other theory by which a
comet can go over fo great a part of the heavens with a re«
gular motion. The motion of this comet was very unequable ^
for about the 20th of Nov, it defcribed about 5^ a day. Then
its motion being retarded between Nor. 9,6 and Dec. 12> to
wit^ in the fpace of I5i days, it defcribed only 40®. But the
motion thereof being afterwards accelerated, it defcribed near
5® a day, till its motion began to be again retarded. And the
theory which juftly correfponds with a motion fo unequable,
and through fo great a part of the heavens, which obferves
the fame laws with the theory of the planets, and which ac-
curately agrees with accurate afironomical obfervations, can*
not be otberwife than true.
And, thinking it would*not be improper, I have given (PI.
18) a true reprefentaiion of the orbit which this comet de-
fcribed, and of the tail which it emitted in feveral places, in
the annexed figure; protradled in the plane of the trajedlory.
In this fcheme ABC reprefents the trajectory of the comet,
. D the fun, DE the axis of the trajectory, DF the line of the
nodes, GH the interfeCUon of the fphere of the orbis magnia
with, the plane of the trajedtory, I the place of the comet
Nov. 4, jinn. 1680 ; K the place of the fame Nov. 1 1 ; L the
place of the fame Nov. ig ; M its place Dec. 12 ; N its place
Dec. 21 ; O its place Dec. 29 ; P its place Jan. 5 following ;
Q its place Jan. 25 ; R its place Feb. 5 ; S its place Feb. 25 ;
T its place March 5 ; and V its place March Q. In deter-
mining the length of the tail, I made the following obfer-
vations.
Nov. 4 and 6, the tail did not appear; Nov. 1 1, the tail juft
begun to fliew itfelf, but did not appear above | deg. long
through a 10 feet telefcope ; Nov. ]?> the tail was feen by
PontMus more than 15^ long; Nov. 18, in New England,
theiail appeared SO^ long, and direCUy oppofite to the fun,
extending itfelf to the planet Mars, which was then in 19, 9^
54'; Nov. 19, in Maryland, the tail was found 15^ or 20^
long ; Dec. 10 (by the obfervation of Mr. Flamjied), the
•*.•
£84 MATHEMATICAL PftlffCIPLKB JBook IJI.
tail paflVd through the middle of the diftance intercepted be-
tween the tail of the Serpent of Ophiuckut and the ftar i in the
fouth wing of Aquila, and did terminate near the ftars A, m, b;
in Bayer's tables. Therefore the end of the tail was in Vf 19¥^,
with hititude about 34 T north; Dec. 11^ it afcended to the
head of Snghta (Bayer's m, fi), terminating in iif 26° 43',
with latitude 38"" 34' north ; Dec. 12, it paffed through the
middle of Sagitie^ nor did it reach much farther ; terminating
in SS 4^, with latitude 42i^ north nearly. But tbefe things
are to be underftood of the length of the brighter part of the
tail ; for with a more faint light, obferved, too, perhaps, in a
ferener flcy, at Rome, Dec. 1% 5^. 40*, by the obfierration of
Ponikaus, the tail arofe to 10^ above the rump of the Swam
and the fide thereof towards the weft and towarda the north
was 4o diftant from this ilar. But about that time die tail was
3^ broad towards the upper end ; and therefore the middle
thereof was 2^ 1.5' diitant from that ftar towards the fouth^
and the upper end wbs >f in 122% with latitude 6l^ north ;
and thence the tail was about 70^ long ; Dec. 21, it extended
almoft to Cajfiopcid's chair, equally diflant from $ and from
Schedir, fo as its diftance from either of the two was equal
to the diftance of the one from the other, and therefore did
terminate in v 24°, with latitude 47^ ; Dec. 29^ it reached
to a contadl with Scheat on its left^ and exadly filled up the
fpace between the two ftars in the northern foot of Andro^
meda, being 54° in length ; and therefore terminated in ^
19°, with 35° of latitude ; Jan. 5, it touched the ftar t in the
breaftof jindronntda on its right fide, and the ftar /u>of the
girdle on its left ; and, according to our obfervations, was 40^
long; butit was curved, and the convex fide thereof lay to
the fouth ; and near the head of the comet it made an anerle
of 4° with the circle which paiTed through the fun and the
comet's head ; but towards the other end it was inclined to
that circle in an angle of about 10° or 11°; and the chord of
the tail contained with that circle an angle of 8°. Jan. 15,
the tail terminated between Alamech and Al^ol, with a light
that was fenfible enough ; but with a faint light it ended over
againft the ftar k in Perfius's fide. The difi^ance of the end
j°/a/t>:K^7my^m .
JM<1£_28^.
i'
li
•«to
4
• I
i»y-
-»
J.
f
ii^
*
Book IH, Of NATURAL PHILOSOPHY. 9B5
of the tail from the circle paffing through the fun and the
comet was 3^ 50' ; and the inclination of the chord of the tail
to that circle was 8l^. Jan. 25 and 26, it (hone with a faint
light to the length of & or 7^ ; and for a night or two after^
when there was a very clear (ky, it extended to the length of
12°, or foraething more, with a light that was very faint and
very hardly to be feen ; hut the axis thereof was exa6Uy di-
re&ed to the bright ftar in the eaftern fhoulder of Auriga,
and therefore deviated from the oppofition of the fun^ to-
wards the north by an angle of 10°. Laftly, Feb, 10, with
a telefcope I obferved the tail 2° long ; for that fainter light
which I fpoke of did not appear through the glaffes. But
Pontham writes, that, on Feb. 7, he faw the tail 12° long.
Feb. 25, the comet was without a tail, and fo continued till
it difappeared.
Now if one refle6ls upon the orbit defcribed, and duly con-
fiders the other appearances of this comet, he will be eafily
fatisfied that the bodies of comets are folid, compa6):, fixed,
and durable, like the bodies of the planets ; for if they were
nothing elfe but the vapours or exhalations of the earth, of
the fun, and other planets, this comet, in its paflage by. the
neighbourhood of the fun, would have been immediately diffi*
pated ; for the heat of the fun is as the denfity of its rays, that
is, reciprocally as the fquare of the diftance of the places
irom the fun. Therefore, fince, on Dec. 8, when the comet
was in its perihelion, the diftance thereof from the centre of
the fun was to the diftance of the earth from the fame as
about 6 to 1000, the fun's heat on the comet was at that time
to the heat of the fummer-fun with us as 1000000 to 96, or
as 28000 to 1 . But the heat of boiling water is about 3 times
greater than the heat which dry earth acquires from the fum-
mer-fun, as I have tried ; and the heat of red-hot iron (if my
coDJe6lure is right) is about three or four times greater than
the heat of boiling water. And therefore the heat which dry
earth on the comet, while in its perihelion, might have con-
ceived from the rays of the fun, was about 2000 times ^eater
than the b^at of ied4iot kon. But by fo fierce a 'heat^ va^
286 MATHEMATICAL PRINCIPLES Book III.
pours 'and exhalations^ and every volatile matter^ muft have
been immediately con i timed and diffipated.
.This cornet^ therefore^ muft have conceived an immenfe
heat from the fun^ and retained that heat for an exceeding long
time 'j for a globe of iron of an inch in diameter^ expofed red-
bot to the open air, will fcarcely lofe all its heat in an hour's
time; but a greater globe would retain its heat longer in the
proportion of its diameter^ becaufe the furface (in proportion
to which it is cooled by the conta6l of the ambient air) is in
that proportion lefs in refpe& of the quantity of the included
hot matter ; and therefore a globe of red hot iron equal to our
earthy that is^ about 40000000 feet in diameter^ would fcarcely
cool in an equal number of days^ or in above 50000 years.
£ut I fufpe6l that the duration of heat may> on account of
fome latent caufes^ increafe in a yet lefs proportion than that
of the diameter ; and I fhould be glad that the true proportion
was inveftigated by experiments,
»It is farther to be obferved^ that the comet in the month
of December, juft after it had been heated by the fun^ did
emit a much longer tail^ and much more fplendid^ than in
the month of Novefnber before^ when it had not yet arrived at
its perihelion ; and, univerfally, the greateft and moft fulgent
tails always arife from comets immediately after their paffing
by the neighbourhood of the fun. Therefore the heat re-
ceived by the comet conduces to the greatnefs of the tail :
from whence, I think^ 1 may infer, that the tail is nothing
clfe but a very fine vapour, which the head or nucleus of the
comet emits by its heat. *
But we have had three feveral opinions about the tails of
comets ; for fome will have it that they are nothing elfe but
the beams of the fun's light tranfmitted through the comets'
heads, which they fuppofe to be tranfparent; others, that
they proceed from the refradlion which light fuffers in pafling
from the comet's head to the earth ; and, laftly, others, that
they are a fort of clouds or vapour conftantly rifing from the
comets' heads, and tending towards the parts oppoiite to the
fun. The firft is the opinion of fuch as are yet unacquainted
with optics ; for the beams of the fun are feen in a darkened
Book III. OP JJATURAL PHILOSOPHY. 287
room only in confequence of the light that is refleAed from
them by the little particles of duft and fmoke which are al-
ways flying about in the air ; and^ for that reafon^ in air ini-
pregnated with thick fmoke, thofe beams appear with great
brightnefs, and move the fenfe vigoroufly ; in a yet finer air
they appear more faint, and are lefs eafily clifcerned ; but in
the heavens, where there is no matter to refle6l the light, they
can never be feen at all. Light is not feen as it is in the
beam, but as it is thence reflefted to our eyes ; for vifion can
be no otherwife produced than by rays falling upon the eyes ;
and, therefore, there muft be fome refle6ling matter in thofe
parts where the tails of the comets are feen : for otherwife,
fince all the celefiial fpaces are equally illuminated by the
fun's light, no part of the heavens could appear with more
fplendbr than another. The fecond opinion is liable to many
difficulties. The tails of comets are never feen variegated
with thofe colours which commonly are infeparable from re-
fradion ; and the diftin6l tranfmiilion of the light of the fixed
fiars and planets to us is a demonftration that the aether or
celefi;ial medium is not endowed with any refra6Hve power :
for as to what is alledged, that the fixed liars have been fome-
times feen by the Egyptians environed with a Coma, or Ca-
pitlitiuniy becaufe that has but rarely happened, it is rath^
to be afcribed to a cafual refra6lion of clouds ; and fo the ra-
diation and fcintillation of the fixed fiars to the refradlions
both of the eyes and air ; for upon laying a telefcope to the
eye, thofe radiations and fcintillations immediately difappear.
By the tremulous agitation of the air and afcending vapours.
It happens that the rays of light are alternately turned afide
fit>m the narrow fpace of the pupil of the eye ; but no fuch
thing can have place in the much wider aperture of the ob-
j^'gl^fs of a telefcope ; and hence it is that a fcintillation is
•occafioned in the former cafe, which ceafes in the latter ; and
this cefiTation in the latter cafe is a demonfi:raUon of the re-
gular tranfmifBon of light through the heavens, without any
fenfible refra&ion. Bdt, to obviate an obje&ion that may be
'made- from the appearing of no tail in fuch comets «8 ihine
but with a faint light, a& if the fecondary rays were then too
1»88 MATHEMATICAL FAIKCIPLES JSook IIL
weak to afieA the eyes, and for that reafon it is that the taih
of the fixed ftars do not appear^ we are to confider^ that bj
the means of telefcopes the light of the fixed ibuB maj be
augmented above an hundred fold^ and yet no tails are feen ;
that the hght of the planets is yet more copious without any
tail; but that comets are feen fometimes with huge tails^
when the light of their heads is but faint and dull. For fo it
happened in the comet of the year 16S0^ when in the month
of December it was fcarcely equal in light to the ftars of the
feeond magnitude^ and yet emitted a notable tail^ extending
to the length of 4(f , 5(f, 60% or 70% and upwards ; and after-
wards^ on the 27 th and 28th of January, when the head ap-
peared but as a ftar of the 7 th magnitude^ yet the tail (as
we faid above), with a light that was fenfible enough^ thou^
faint^ was ftretched out to 6 or 7 degrees in length, and with
a languiihing light that was more difficultly feen, even to 19!^,
and upwards. But on the 9th and 10th of February, when
to the naked eye the head appeared no more, through a tele*
fcope I viewed the tail of 2° in length. But farther ; if the tail
was owing to the refradUon of the celellial matter, and did
deviate from the oppofition of the fun, according to the figure
of the heavens, that deviation in the fame places of the
heavens fhould be always dire6led towards the fame parts.
But the comet of the year I68O, December 28**. Sl*". P. M.
at London, was feen in K 8^ 41', with latitude north 28^ 6';
while the fun was in lif 18® 26'. And the comet of the year
1577, December 29**. was in K 8® 41', with latitude north 28^
40', and the fun, as before, in about icP 18® 26'. In both
cafes the fituation of the earth was the fame, and the comet
appeared in the fame place of the heavens ; yet in the former
cafe the tail of the comet (as well by my obfervations as by
the obfervations of others) deviated from the oppo&tion of
the fun towards the north by an angle of 4| degrees ; where*
as in the latter there was (according to the obfervations of
Tycho) a deviation of 21 degrees towards the foutb. The
refra^on, therefore, of the heavens being thus difproved, it
remains that the phenomena of the tails of comets muft be
derived from fome refledling matter.
Book III. ' OF NATURAL PHILOSOPHY. . S89
And that the tails of comets do arife from their heads^ and
tend towards the parts oppofite to the fun, is farther confirm-
ed from the laws which the tails obferve. As that, lying in
the planes of the comets' orbits which pafs through the fun,
they conftantly deviate from the oppofition of the fun to-
wards the parts which the comets' heads in their progrefs
along thefe orbits have left. That to a fpedator, placed in
thofe planes, they appear in the parts dire6lly oppofite to the
fun ; but, as the fpedlator recedes from thofe planes^ their de-
viation begins to appear, and daily becomes greater. That
the deviation, cateris paribus^ appelars lefs when the tail is
more obliquje to the orbit of the comet, as well as when the
head of the comet approaches nearer to the fun, efpecially if
the angle of deviation is eftimated near the head of the
comet. That the tails which have no deviation appear
ftraight, but the tails which deviate are likewife bended into
a certain curvature. That this curvature is greater when the
deviation is greater; and is more fenfible when the tail, ^
cateris paribus, is longer ; for in the ihorter tails the curva-*
ture is hardly to be perceived. That the angle of deviation is
lefs neat the comet's head, but greater towards the other end
of the tail; and that becaufe the convex fide of the tail re-
gards the parts from which the deviation is made, and which
lie in a right line drawn out infinitely from the fun jfbrough
the comet^ head. And that the tails that are long and broad,
and fliine with a fi;ronger light, appear more refplendeiit and
more exa6Uy defined on the convex than on the concave
fide. Upon which accounts it is plain that the p/uenomena
of the tails of comets depend upon the motions of their
heads, and by no means upon the places of the hiSavens in
which their heads are feen ; and that, therefore, the tails of
comets do not proceed from the refra6lion of the heavens,
but from their own heads, which frirnifh the mattef that
fdrms the tail. For, as in our air, the fmoke of a heated body
afcends either perpendicularly if the body is at reft, or oblique-
ly if the body is moved obliquely, fo in the heavens, where
all bodies gravitate towards die fun, fmoke and vapour muA ^^
(as we have already faid) afcend from the fun^ and either rife
Vox. 11. V ' '
tgo BCikTH£MATicAL rmiNciPLK» Book III
perpendicalarly if the fmokiog body b at rt&, or obliqaely if
the body^ in all the progrefs of iU motion, is always leaving
thofe places from which the upper or higher parts of the
vapour hud rifen before; and that obliquity will be leaft
where the vapour afcends with mod velocity, to wit, near the
finoking body, when that is near the fun. But, becaufe the
obliquity varies, the column of vapour will be incurvated;
and becaufe the vapour in the preceding fides is fomething
more recent, that is, has afcended fomething more late
firom the body, it will therefore be fomething more denfe on
that fide, and muft on that account refleA more light, as well
as be better defined. I add nothing concerning the fudden.
uncertain agitation of the tails of comets, and their irregular
figures, which authors fometimes defcribe^ becaufe they
may arife from ^e mutations of our air^ and the motions
of our clouds, in part obfcuring thofe tails ; or^ perhaps, fiiom
parts of the Fia La£iea, which might have been confounded
with and miflaken for parts of the tails of the comets as they
' pafied by.
But that the atmofpheres of comets may fumilh a fupply
of vapour great enough to fill fo immenfe (paces^ we may
eafily underftand from the rarity of our own air ; for the air
near the furface of our earth poflefTes a fpace 8oO times
greater than water of the fame weight ; and therefore a cylin-
der of air 850 feet high is of equal weight with a cylinder of
water of the fame breadth, and but one foot high. But a
cylinder of air reaching to the top of the atmofphere is of
equal weight with a cylinder of water about 33 feet high :
and, therefore, if from the whole cylinder of air the lower
part of 850 feet high is taken away, the remaining upper
part will be of equal weight with a cylinder of water 32 feet
high : and from thence (and by the hypothefis, confirmed by
many. experiments, that the compreilion of air is as the.
weight of the incumbent atmofphere, and that the force of
gravity is reciprocally as the fquare of the diftance from the
centre of the earth) raifing a calculus, by cor. prop. 22, book
^, I found, that, at the height of one femi-diameter of the
earth, reckoned from the earth's furface, the air is more rare
than w;ith us in a far greater proportion than of the whok
Sdok lU. OF NATURAL PHILOSOPHfT*' QQl
fpace within the orb of Saturn to a fpherical fpace of one
inch in diameter ; and therefore if a fphere of our air of but
pne inch in thicknefs was equally rarefied with the air
at the height of one femi^diameter of the earth from the
earth's furface, it would fill all the regions of the planets to
the orb of Saturn, and far beyond it. Wherefore fince the
air at greater diftances is immenfely rarefied, and the coma or
atmofphere of comets is ordinarily about ten times higher,
reckoning from their centres, than the furface of the nucleus^p
and the taib rife yet higher, they mull therefore be exceed-
ingly rare : and though, on account of the much thicker b,U
mofpheres of comets, and the great gravitation of their
bodies towards the fun, as well as of the particles of their air
and vapours mutually one towards another, it may happen that
the air in the celefi;ial fpaces and in the tais 6f comets is fiot ^
fo vafily rarefied, yet from this computation it is plain that a
very fmall quantity of air and vapour is abundantly fufficient
to produce all the appearances of the tails of comets ; for ^
that they are, indeed, of a very notable rarity appears from
the fhining of the liars through them. The atmofphere of
the earth, illuminated by the fun's light, though but of a few
miles in thicknefs, quiteobfcures andextinguifhes thelight not
only of all the ft^n^but even of the moon itfelf ; whereas Ae
fmallell liars are fe^ to Ihine through the immenfe i;^ickne&
of the tails of comets, likewife illuminated by the fun, witn-
ont the leall diminution of their fplendor. Nor is the bright-
nefs of the tails of moll comets ordinarily greater than that
of Our air, an inch or two in thicknefs, reHedling in a darkened
room thelight of the fun-beams let in by a faltfe of the
window-lhutter. ^ "lit'
And we may pretty nearly determine the time fpent dur-^
ing the afcent of the vapour from the comet^head to the ex-
tremity of the tail, by drawing a right line from theex#bmil^
of the tail to the fun, and marking the place where that right
line interfe6ls the comet's orbit ; for the vapour that is noW
in the extremity of the tail, if it has afcended in a right line ^^
from die fun^ mnft hove begun to rife from the head at the ^^^
^lime when the head was in the point of interfedUon. It is
fQ2 MATHEMATICAL PftlNCIPLSS fioofc HI.
true, the vapour does not rife in a right line from the fun,
but, retainipg the motion which it had from the comet before
iU alceuti and compounding that motion with its motion of
afcent, anfes obliquely ; and, therefore, the folution of the
problem will be more exai&, if we draw the line which inter*
fe6U tlie orbit parallel to the length of the tail ; or rather
(becaufe of the curvilinear motion of the comet) diverging a
little from the line or length of the tail. And by means of
this principle I found that the vapour which, January 25,
was in the extremity of the tail, had begun to rife from the
head before December 11, and therefore had fpent in its
whole afcent 45 days ; but that the whole tail which ap-
peared on December 10 had finifhed its afcent in the
fpace of the two days then elapfed from the time of the
corners being in its perihelion. The vapour, therefore, about
the beginning and in the neighbourhood of the fun rofe with
the greaieil velucity, and afterwards continued to afcend with
a motion conilantly retarded by its own gravity; and the
" higher it afcended, the more it added to the length of the
tail ; and while the tail continued to be feen, it was made up
of almoft all that vapour which had rifen fince the time of
the comet's being in its perihelion ; nor did that part of the
vaf>our which had rifen firft, and which formed the extremity
of the tail, ceafe to appear, till its too great diftance, as well
from the fun, from which it received its light, as from our eyes,
rendered it invifible. Whence alfo it is that the tails of
other comets which are ihort do not rife from their heads
with a fwift and continual motion, and foon after difappear,
but are permanent and lading columns of vapours and ex«
halations^ which, afcending from the heads with a flow mo«
tion of many days, and partaking of the motion of the heads
which they had from the beginning, continue to go along
together with them through the heavens. From whence
again we have another Argument proving the celeftial fpaces
to be free, and without refiftance, fince in them not only the
folid bodies of the planets and comets, but alfo the extremely
rare vapours of comets' tails, maintmn their rapid motions
with great freedom, and for an exceeding long tiinie<
Book in. OP NATURAL liHlXOSOPHY. 293
Kepler afcribes the afcent of the tails of the comets to the
atmofpheres of their heads ; and their direftion towards the
parts oppofite to the fun to the a6):ion of the rays of light
carrying along with them the matter of the comets' tails ; and
without any great incongruity we may fuppofe, that, in fo
free fpaces, fo fine a matter as that of the aether may yield to
the a6lion of the rays of the fun's light, though thofe rays
are not able feniibly to move the grofs fubftances in our parts,
which are clogged with fo palpable a refiftance. Another
author thinks that there may be a fort of particles of matter
endowed with a principle of levity, as well as others are with
a power of gravity; that the matter of the tails of comets may
be of the former fort, and that its afcent from the fun. may be
owing to its levity ; but, confidering that the gravity of ter-
reftrial bodies is as the matter of the bodies, and therefore can
be neither more nor lefs in the fame quantity of matter, I am
inclined to believe that this afcent may rather proceed from
the rarefadlion of the matter of the comets' tails. The afcent
of fmoke in a chimney is owing to the impulfe of the air with
which it is entangled. The air rarefied by heat afcends, be-
caufe its fpecific gravity is diminifhed, and in its afcent
carries along with it the fmoke with which it is engaged ; and
why may not the tail of a comet rife from the fun ofier the
fhme manner? For the fun's rays do not a&, upon the me-
dinms which they pervade otherwife than by refle&ion and
refra6lion; and thofe refledling particles heated by this ac-
tion, heat the. matter of the aether which is invoked with
them. That matter is rarefied by the heat which iTacquires,
and becaufe, by this rarefadion, the fpecific gravity with
which it tended towards the fun before is diminifhed, it will
afcend therefrom, and carry along with it the refiedling par-
ticles of which the tail of the comet is compofed. But the
afcent of the vapours is further promoted by their circum-
gyration about thefon,in confeqaence whereof they endeavour
to recede from the fun, while the fun's atmofphere and the
other matter of the heavens are either altogether quiefcent,
or are only moved with a flower circumgyration derived from
the rotaticA of the fun. And thefe are the caufes of the
US
294 MAtHEMATiCAL PRINCIPLES Book III,
afcent of the tails of the comets in tbe^ neighbourhood of
the fun, where their orbits are bent into a greater curvature^
and the comets themfelves are plunged into the denfer and
therefore heavier parts of the fun's atmofphere: upon which
account they do then emit tails of an huge length; for tile
tails which then arife^ retaining their own proper motion^ and
in the mean time gravitating towards the fun^ mull be revolv^
ed in ellipfes about the fun in like manner as the heads are^
and by that motion mud always accompany the heads^ and
freely adhere to them. For the gravitation of the vapours
towards the fun can no more force the tails to abandon the
r
heads^ and defcend to th^ fun^ than the gravitation of the
beads can oblige them to fall from the tails. They muft by
their common gravity either fall together towards the fun,
or be retarded together in their common afcent thierefrom ;
and, therefore (whether from the caufes already defcribed,
or from any others), the tails and heads of comets may eafily
acquire and freely retain any poiition one to the other> with-
out difturbance or impediment from that common gravita-
tion.
The tails, therefore, that rife in the perihelion pofitions
of the comets will go along with their heads into far remote
parts, and together with the heads will either return again
from thence to us, after a long courfe of years, or rather will
be there rarefied, and by degrees quite vanifh away; for after-
wards, in the defcent of the heads towards the fun, new fliort
tails will he emitted from the heads with a flow motion ; and
thofe tails by degrees will be augmented immenfely, efpecially
in fuch comets as in their perihelion diflances defcend as low
as the fun's atmofphere; for all vapour in thofe free.fpaces
is in a perpetual ftate of rarefadion and dilatation; and from
hence it is that the tails of all comets are broader at their
upper extremity than near their heads. And it is notunlikely but
that the vapour, thus perpetually rarefied and dilated, may
be at lafi: diffipated and fcattered through the whole heavens^
and by little and little be attradled towards the planets by
its gravity, and mixed with their atmofphere ; for as the feas
are abfolutely neceffary to the conftitutipn of ou;r earl^., th^t
Book III. OP NATUHAL PHILOSOPHY. £95
%
I
from them^ the fun, by its heat, may exhale a fufficient quan-
tity of vapours, which, being gathered together into clouds,
may drop down in rain, for watering of the earth, and for
the produdiion and nourifhment of vegetables; or, being con-
den fed with cold on the tops of mountains (as fomephilofo-
phers with reafon judge), may run down in fprings and rivers ;
fo for the confervation of the feas, and fluids of the planets,
comets feem to be required, that, from their exhalations
and vapours condenfed, the waftes of the planetary fluids
fpent upon vegetation and putrefadlion, and converted into
dry earth, may be continually fupplied and made up ; for all
vegetables entirely derive their growths from fluids, and after-
wards, in great meafure, are turned into dry earth by putrefac*
tion; and a fort of flime is always found to fettle at the hot"
torn of putrefied fluids; and hence it is that the bulk of the
folid earth is continually increafed;. and the fluids, if they are
not fupplied from without, mufl; be in a continual decreafe,
and quite fail at lafl:. I fufpecft, moreover, that it is chiefljr
from the comets that fpirit comes, which is, indeed the fmall-
efl; but the mofl, fubtle and ufeful part of our air, and fo much
required to fuftain the life of all things with us.
The atmofpheres of comets, in their defcent towards the
fun, by running out into the tails, are fpent and diminiflied,
and become narrower, at leafl: on that fide which regards the
fun; and in receding from the fun, when they lefs run out
into the tails, they are again enlarged, if Hevdiu$ has juftly
marked their appearances. But they are feen Jeaft of all
jufl; after they have been mofl; heated by the fun, and on that
account then emit the longefl; and mofl; refplendent tails ; and,
perhaps, at the fame time, the nuclei are environed with a
denfer and blacker fmoke in the lowermoft parts of their at-
mofphere ; for Ifaioke that is raifed by a great and Intenfe
heat is commonly the denfer and blacker. Thus the head of
that comet 'which we have been defcribing, at equal dif-
tances both from the fun apd from the earth, appeared darker
after it had pafled by its perihelion than it did before ; for in .
the month of December It was commonly compared w ith th^
ilars of the third magnitade, but in November witlTtbofie of the
U4
(2g6 MATHEMATICAL PElNCIPLfi^ Book III.
firfc or fccond ; and fuch as faw both appearances have defcrib"
ed the firft as of another and greater comet than the fecood.
For, November 19^ this comet appeared to a young man at
CamlfriJge, though with a pale and dull lights jet equal to
Spica rirgifiis ; and at that time it (hone with greater bright-
nefs than it did afterwards. And Montenari, November 9J0,
ft. vet. obferved it larger than the flars of the firft magnitude^
its tail being then 2 degrees long. And Mr. Storer (by let-
ters which have come into my hands) writes, that in the
month of December, when the tail appeared of the greateft
bulk and fpleudor, the head was but fmall, and far lefs than
that which was feen in the month of November before fun-
rifing ; and> conje<5luring at the caufe of the appearance, he
judged it to proceed from there being a greater quantity of
matter in the head at firft, which was afterwards gradual-
ly fpent.
And, which farther makes for the fame purpofe, I find^ that
the heads of other comets, which did put forth tails of the
greateft bulk and fplendor, have appeared but pbfcure and
fmall. For in Brafil, March 5, 1 668, 7^ P. M., St. N. P.
Valentinm EJlancius faw a comet near the horizon, and
towards the fouth weft, with a head fo fmall as fcarcely to
be difcerned, but with a tail above meafure fplendid, fo that
the refle6iion thereof from the fea was eafily feen by thofe
who ftood upon the fliore; and it looked like a fiery
beam*extended 23° in length from the weft to fouth, almoft
parallel to the horizon. But this exceffive fplendor continued
only three days, decreafiug apace afterwards; and while
the fplendor was decreafing, the bulk of the tail increafed :
whence in Portugal it is faid to have taken up one quarter
of the heavens, that is, 45 degrees, extending from weft to eaft
with a very notable fplendor, though the whole tail was not
feen in thofe parts, becaufe the head was always hid under
the horizon : and from the increafe of the bulk and decreafe
of the fplendor of the tail, it appears that the head was then
in its recefs from the fun, and had been very near to it in its
perihelion, as the comet of 1680 was^ And we read, in the
Saxon Chxonich, of a like comet appearing in the year 1106,
Book III. OF NATUKAL PHILOSOPHY. Qffl,
ihtftar whereof was fmall and obfcure (as tbat^of 1680), bui.
thejplendour of its tail was very bright, and like a hugejiery
beamftreiched out in a direSion between the ettji and north, a«
Hevelius has it alfo irom Simean, the monk of Durham.
This comet appeared in the beginning of February, about the
evening, and towards the fouth weft part of heaven ; from
whence^ and from the pofition of the tail, we infer that the
head was near the fun. Matthew Paris fays, Jt weis diftant
from the fun by about a cubit, from three of the clock^tdXhev fix)
till nine, putting forth a long tail. Such alfo was that moft
refplendent comet defcribed by Arijiotle, Hb. 1. Meteor. 6.
The head whereof could not be feen, becaufe it hadfet before
the fun, or at leajt was hid under the fun's rays ; but next day
it was feen as well as might be; for, having left the fun but a
very little way, itfet immediately after it. And tlit fcattered
light of the head, obfcured by the too great fplendour (of the
tail) did not yet appear. But afterwards (as Arijiotle fays)
when the fplendour (of the tail) was now diminifhed (the head
of), the comet recovered its native brightnefs; and thefplendor
{pi its tail) reached now to a third part of the heavens (that is^
to 60°). This appearance wa^ in the winter feafon (an. 4,
Olymp. \0\),and, rifing to Orion's girdle,it therevanijhedaway.
It is true that the comet of l6l8, which came out diredllj
from under the fun's rays with a very large tail, feemed to
equal, if not to exceed, the ftars of the firft magnitude ; but,
then, abundance of other comets have appeared yet greater
than this, that put forth fhorter tails ; fome of which are faid
to have appeared as big as Jupiter, others as big as Venus, or
even as the moon.
We have faid, that comets are a fort of planets revolved
in very eccentric orbits about the fun ; and as, in the planets
which are without tails, thofe are commonly leis which are
revolved in leffer orbits, and nearer to the fun, fo in comets
it is probable that thofe which in their perihelion approach
nearer to the fun are generally of lefs magnitude, that they
may not agitate the fun too mucM by their attractions. . ^ut
as to the tranfverfe diameters of their orbits^ a^d the peria^l^
times of their revdiutioii3j I leave them to be 4e^nn{^ ^
198 XATHEMATICAL PBIITCIPLES BookllL
oompariiig cooiets together which after long intervals of time
letom again in the fame orbit. In the mean time, the fol*
lowing propofilioB mar give fome light in that enquiry.
PROPOSITION XLII. PROBLEM XXII.
To camQ a camtCt trajeSory found as above.
Opebation 1. Aflame that pofition of the plane of the
IrajeAory which was determined according to the preceding
ptopofition ; andfele^l three placet of the comet^dedaced from
very accurate obfervations^ and at great diftanees one from
the other. Then fuppofe A to repiefent the time between
the 6rft obfervation and the fecond, and B the time between
the fecond and the third ; but it will be convenient that in
one of thofe times the comet be in its perigeon^ or at lead
not far from it. From thofe apparent places find^ by trigo-'
nometric operations, the three true places of the comet in
that aflumed plane of the traje<^oiy; then through the
places found, and about the centre of the fun as the focus,
defcribc a conic fedlion by arithmetical operations, according
to prop. '21, book 1. Let the areas of this figure which are
terminated by radii drawn from the fun to the places found
be D and E ; to wit, D the area between the firft obfervation
and the fecond, and E the .'jrea between the fecond and
third ; and let T reprefent the whole time in which the
whole area D + E fliould be defcribed ^ith the velocity of
the comet found by prop. l6, book 1.
Oper. 2. Retaining the inclination of the plane of the
trajeftory to the plane of the ecliptic, let the longitude of the
nodes of the plane of the trajeftory be increafed by the addi-
tion of 120 or 30 minutes, which call P. Then from the afore-
faid three obferved places of the comet let the three true
places be found (as before) in this new plane; as alfo the orbit
paffing through thofe places, and the two areas of the fame
defcribed between the two obfervations, which call d and e ;
and let t be the whole time in which the whole area d + e
(hould be defcribed.
Opeb. S. Retaining the longitude of the nodes in the firil
operation, let the inclination of the plane of the trajectory to
the plane of the ecliptic be increafed by adding thereto 90' or
Voffk in. OF NATURAL PHILOSOPHY* . ^ £9^
50'> which call Q. Then from the aforefaid three obferved ap-
parent places of the comet let the three true places be found
in this new plane^ as well as the orbit paffing through them,
and the two areas of the fame defcribed between the obfer->
vation, which call i and c ; and let r be the whole time in
which the whole area S + e ihould be defcribed^
Then taking C to 1 as A to B; and G to 1 as D to E;
and g to 1 as d to e ; and y to 1 as 2 to c; let S be the tme
time between the firft obfervation and the third ; and^
obferving well the figns + and — , let fuch numbers m and n
be found out as wiU make 2G — 2C, = mG — mg + nG
— ny; and 2T — 2S = mT — mt + nT — nr. And if,
in. the firft operation, I repreients the inclination of the plane
of the traje6lory to the plane of the ecliptic, and K the longir
tude of either node, then I + nQ will be the true inclination
of the plane of the trajedlory to the plane of the ecliptic, and
K + mP the (rue longitude of the node. And, lafUy, if in
the firft, fecond, and third operations, the quantities R, r, and
^, reprefent ' the parameters of the trajedtory, and the quanti-
ties -T-j ^, -f the tranfverfe diameters of the fame, then R+
Li 1 A
mr — mR + nj — nR will be the true parameter, and
? — ; — , T . — : r will be the true tranfverfe dia-
li + Hii — mil + nX — nil
meter of tl^e traje^ry which the comet defcribes; and from the
tranfverfe diameter given the periodic time of the comet is alfo
given. Q.E.I. But the periodic times of the revolutions of
comets, and the tranfverfe diameters of their orbits, cannot be
accurately enough determined but by comparing comets toge-
ther which appear at different times. If, after equal intervals
of time, feveral comets are found to have defcribed the fame
orbit, we may thence conclude that they are all but one and
the fame comet revolved in the fame orbit ; and then from the
times of their revolutions the tranfverfe diameters of their orbits
will be given, and from thofe diameters the elliptic orbits
themfelves will be determined.
To this parpofe the trajectories of many comets ought to
be computed, foppofing thofe trajectories to be parabolic;
(
SOO MATBBMATICAL FBHtCIPLES Book IIL
Ibr fttch trafoAories will dwajs Dearly Bgree with tbe/iihriio-
ttfiM, as appemrs not only fixMn the parabolic tngedoiy of
the comet of Ibe year 1680, which I compared above with
the obfervatioiis^ but Ukewife fixMn that of the notable comet
which appeared ia the years 1664 and l665» and was obfenred
by Htvtlims, who, from his own obferYations, caknilated the
loagitodes and latitudes tbeieof, though with little accuracy.
But from the fiime obienrations Dr. HaUty did again com*
pate its places : and from thofe new places determined its
traje^lory, finding its afceoding node ia n (!l^ IS* 55"; the
indination of the orbit to the plane of the ecliptic !2l° IS' 4Ql';
the diftance of its perihelioB from the node, eftimafeed in the
comet's orbit, -tg^ 97' SCT, iu perihelion in a 8^ 40' 30",
with heliocentric latitude footh {& 01' 45' ; the comet to
have been in ito periiielion X^amber 24^. 1 \K ^ P. M.
equal time at Ixmdom, or 13^. d' at DmUzuJt, O. S. ; and that
the htm rtdum of the parabola was 410S8fS fnch parts as the
fan's mean diftance from the earth is fnppofed to contain
100000. And how nearly the places of the comet computed
in this orbit agree with the obfenrationsi, will sppear from the
annexed table, calculated by Dr. HalUj.
»
KD
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Ook III. OF NATTTRAL PHILOSOPHY. 301
In February, the beginning of the j'ear 1665, the firft ftar of
rie», which 1 {hall hereafter call y, was in qr 28"* 30' 15^
ith T^ 8' 68" north lat. ; the fecond ftar of Aries was in V ^fiP
T 18", with 8^ 28' 16" north lat. ; and another ftar of the
eviMrth magnitude, which I call A, was in V QS^ 24' 45",
jrith 8^ 28' S3" north lat. The . comet Feb. 7*. 7*^. 30' at
Ptfri« (that is, Feb. 7K 8*». 37' at Dantzick) O. S. made a
tiiaifigle with thofe fiars y and A, which was right-angled in
y ; and the diftance of the comet from the ftar, y was eqa^al
*o the diftance of the ftars y and A, that is, 1° 19' 46" of a
great circle; and therefore in the parallel of the latitude of
the ftar y it was 1° 20' 26". Therefore if from the longitude
of the ftar y there be fubduded the longitude 1® 20' 26", there
will remain the longitude of the comet *Y» 27^ 9* 49". M.
Auzout, from this obfervation of his, placed the comet in ***
fi7*>0', nearly; and, by the fcheme in which Dr. Hooke de-
lineated its motion, it was then in <r 26° 59' 24". I place
it in V 27** 4' 46", taking the middle between the two ex-
tremes.
From the fame obfervation, M. Auzout made the latitude
of the comet at that time 7® and 4' or 5' to the north; but
ihe had done better to have made it 7° 3' 29", the difierence
of the latitudes of the comet and the ftar y being equal to the
difference of the longitude of the ftars y and A.
February 22**. 7^. 30' at London, that is, February 22**. 8*.
46' at Dantzick, the diftance of the comet from the ftar A,
according to Dr. Hooke's obfervation, as was delineated bv
himfelf in a fcheme, and alfo by the obfervations of M . Au^
zout, delineated in like manner by M. Petit, was a fifth part
of the diftance between the ftar A and the firft ftar of Aries,
or 15' 57" ; and the diflxince of the &>met from a right line
joining the ftar A and the firft of Aries was a fourth part of
the fame fifth part, that is, 4' ; and therefore the comet was in
qr 28*" 29' 46", with &" 12' 36" north lat.
March 1, 7*^. 0* at London, that is, March 1, 8**. l6' at
Dantzick, the comet was obferved near the fecond ftar in Aries,
the diftance between them being to the diftance between the
firft and fecond fliars ia Aries^ that is, to I'' 33', as 4 to 45
I
%
90S MATflBltATICAL V«IMC1F1.BS Book Uf*
sceofdiDg to Dr. HookCf or as S to 23 according to M. Got"
Hptm. And, tbeiefore, Ibe diftance of the comet firom the
fiecond ftar in Aries was ft 16" according to Dr. Hooke, or 8' 6"
Aoooiding to M. GoUigmia; or, taking a mean between both^
8* 10". Bui, accofding to M. Goitignie$, tbe comet had
gone beyond the fecond ftar of Aries about a fourth or a fifth
part of ibe fpace that it commonly went oYcr in a day, to wit,
about r 36" (in which be agrees veiy well with M. Auzout) ;
or, according to Dr. Hooke^ not quite fo much, as perhaps
only 1'. Wherefore if to the longitude of the firft ftar in Aries
we add 1', and 8' 10" to its latitude, we fliall have the longi-
tude of the comet "V 29"* IS', with 8^ 3ff 26" north lat.
March 7, 7^ Sd at Ptfrii (that is, March 7, 8^. 37' at
Damtzick), from the obfervatioos of M. Auzout, the diftance
€»f tbe cpmet from the fecond ftar in Aries was equal to die dift-
ance of that ffaur from the ftar A, that is, 59! 29" ; and the
difference of the longi^de of the comet and the fecond flar in
Aries was 44' or 46', or, taking a mean quantity, 45' SO" ;
and therefore the comet was in ^ 0^ 2" 48". From the
Icheme of the obfervations of M. Auzout, conftruded by M.
Pitit, Hevtlius colle6led the latitude of the comet 8^ 54'.
But the engraver did not rightly trace the curvature ot th^
comet's way toward the end of the motion ; and Hevclius, in
the fcheme of M. Auzout's obfervations which he conftru6led
himfelf, corre<^d this irregular curvature, and fo made the
latitude of the comet 8** 55' 30". And, by farther corredting
this irregularity, the latitude may become 8^ 56', or 8° 57'.
This comet was alfo feen March g, and at that time
its place muft have been in » 0° 1 8', with 9^ 3i' north lat.
nearly.
This comet appeared three months together, in which fpace
of time it travelled over almoft fix figns, and in one of the
days thereof defcribed almoft 20 deg. Its courfe did very
much deviate from a great circle, bending towards the north,
and its motion towards the end from retrograde became di-
rect ; and, notwithftanding its courfe was fo uncommon, yet
by the table it appears that the theory, from beginning to.
end, agrees with tbe obfervations no lels accurately than the
Book IIL OP NATURAL FH1LOSOPBY. 303
theories of the planets ufually do with the ohfervations of
them ; but we are to fiibdu6l about 2' when the comet wa«
fwifteft, which we may efFe6l by taking off 12" from the angle
between the afcending node and the perihelion^ or by making
that angle 49° 27' 18". The annual parallax of both thefe
comets (this and the preceding) was very confpiciious^ and by
its quantity demonftrates the annual motion of the earth in
the orbis magntis.
This theory is likewife confirmed by the motion of that co-
met, which in the year 1683 appeared retrograde, in an orbit
whofe plane contained almoft a right anglf with the plane of
the ecliptic, and whofe afcending node (by the computation
of Dr. Halley) was in liji 23° 23' ; the inclination of its orbit
to the ecliptic 83° 1 1' ; its perihelion . in n 25^ £9' 30" ; ita
perihelion difiance from the fun 56020 of fuch parts as the
radius of the orbis magnus contains 100000 \ and the time of
its perihelion July 2**. 3**. 50'. And the places thereof, com-
puted by Dr. Halley in this orbit, are compared with the
places of the fame obferved by Mr. Flamjicd, in the following^
table.
MATHEMATICAL PRINCIPLBS Boofc lib,
Si
+ + + I 1 11+++ Mil 11 +
i|
11
|3i|si3g5|iiiii3ii
1!
5 3
z 1
i|
1}
.S:^«BS8SS!8SSS&| 3 = S.
1
4
.tji- d«* ffij 22 2 o 2 22 :i22
t 1
This theory is yet farther confirmed by the motion of that
leliogmde comet which appeared in the year 1688. The
afcendingnodeoflhi!(by Dr. Hii%'s compotaUon) was in
»j 21" Iff 30" ; the inclination of iu orhit to the plane of the
ecliptic 17° iff 00" i il» perihelion in = 2° 58' SO" i its peri-
helion diftancefrom Ihe fon 58388 parts, of which the radios
of the oriis magma contains 100000 ; the eqnal time of the
comet's being in iu perihelion S^t. 4'. 7'. sg-. And iU
places, coUeSed from Mr. nnn^tifs-obfenations, are com-
pared with its places computed from our theory in the follow-
ing table.
Book IIL
OP NATURAL PHILOSOPHY.
305
Diff.
Lat.
- O'^C<»->C0OC0'-iOOO
+ -K+++ 1 ++. 1 + +
Diff.
Long.
1 + 1 1 + + + ++ 1 +
Lat.Nor.
obferved.
5 »'5C^|>s.0»»0t^0^O'-<O»C
"o>c»CNrN — — -< — /—
Com. Long,
obferved.
00 -# CTi'o c* *o d »^ 'o o^ d
Lat. Nor.
comp.
« i^(NO(Nt^C0OQ0G0'>O0
- 0'«f«0'*'«*'«J<0'*0'*»-'
O "* O eo l>.^ CO CO CO ^ Oi
* ^•-^C^OCOO<*-«C^COO<'*
•...•.. . • « • . ••
° <N 0< CN CM -• ^ »^ ^ -H
Comet's
Lpn. comp.
~ OOCO^OCO-^^HCOOVO-^O
^ -^^O t^ C^r^^ O CM O GO rh
^'«i«c0C*C000C0'^j«O»O>'*
• •• • •• ... ■«
^ 00 -«* a>^ o* *o o 'o r^ c^ o
Sun's
place.
5 6 •o ^ CO c* c^^d -^ d »o »o ■
q»ococo<N — — ^»-iod
o hi N. 00 cfi^ r>l o^ oj CO 'o K.
r-l r-l .-* C? 0< C< OJ
1682
App. time.
COCOO»OCI•*COC^*CO^O*
^•-^oooooooob
^c?^d-: CM did ^ 4^o6M
^'^C^CMCNCNCO ^^
^ t
This theory is alfo confirmed by the retrograde motion of
the comet that appeared in the year 1723. The afcending
node of this comet (according to the computation of Mr.
Bradley, Savih'an Profeffor of Aftronomy at Oxford) was in
<r 14° Iff. The inclination of the orbit to the plane of the
ecliptic 49° 59'. Its perihelion was in « 12° 15' 20". Its
perihelion diftance from the fun 998651 parts, of which the
radius of the orbis magnus contains 1000000, and the equal
time of its perihelion September 16<». 16^. 10'. The places of
this comet computed in this orbit by Mr. Bradley, and com-
pared with the places obferved by himfelf, his uncle Mr.
Vol. II. X
506 MATHEMATICAL PEIKC1PLB8 Book III,
Pamd, and Dr. Hmlley, may be feea in the foUowiDg
Uble.
1723.
£q. time.
d h
oa. 9.8. 5
10.6.21
12.7.22
14.8.37
15.6.33
21.6.22
22.6.24
24.8. 2
29.8.56
30.6.20
Nut, 5.5.33
%.7. 6
14.6.20
Conet't
Loog. obf.
:r7.22.13
6.41.12
5.39.58
4.59.49
4.47.41
4. 2.32
3.59. 2
\lkc
20,7.45[
. 7.6.451
3.56.17
3.58. 9
4. 16. JO
4.29.36
5. 2.16
5.42.20
8. 4.13
LatNbr.
obC
•1 « •
5. 2.
7.44.13
11.55.
14.43.50
15.40.51
19.4l-i9
20. %At
3.55.29 20.55.18
Comet't
Loo. con.
Lai. Nor.
CGBipi*
22.20.27
22.32.2
23.3S
24. 4.
24.4
25.24.45
26.54.18
5:7.21.26
6.41.42
5.4a 19
5. 0.37
4.47.45
4. 2.21
3.59.10
3.55.11
3.56.42
3.58.17
4.16.23
Lon.
5. 2.47]+ 491— 47
+ 55
+ 5
—11
7.43.18(— 50
11.54.55—21
14.44. 1—48
15.4a55
19-42. 3 + 11
20. 8.17— 8
20.55. 9 + 18
22.2a 101—25
22.32. 1
23.38. 7
4.29.54 24. 4
5. 2.5l)24.48.l6|--35
5.43. 1 3/25.25. 1 7/— -531— 32
8. 3.53/26.53.42/+ 18/+ 36
Ut.
I
8
7
•18
?
4
—14
— 5
+
+ 17
+ 16
+e6
k-10
+ 30
From thefe examples it is abundantly evident that the mo-
tions of comets are no lefs accurately reprefented by our the-
ory than the motions of the planets commonly are by the the-
ories of them ; and^ therefore^ by means of this theory^ we
may enumerate the orbits of comets, and fo difcover. the pe-
riodic time of a comet's revolution in any orbit ; whence, at
lail, we fhall have the tranfverfe diameters of their elliptic
orbits and their aphelion diftances.
That retrograde comet which appeared in the year I607
defcribed an orbit whofe afcending node (according to Dr.
Halley's computation) was in b 2ff* 21' ; and the inclination
of the plane of the orbit to the plane of the ecliptic 17° 2^ ;
whofe perihelion was in x:: 2° I6' ; and its perihelion diftance
from the fun 58680 of fuch parts as the radius of the orbis
magnus contains 100000 ; and the comet was in its perihelion
O&obtr 16*. 3**. 50'; which orbit agrees very nearly with
the orbit of the comet which was feen in 1682. If tbefe were
not two different comets, but one and the fame^ that comet
will finifh one revolution in the fpace of 75 years ; and the
greater axis of its orbit will be to the greater axi$ of the or6t9
Book III. OF NATURAL PHILOSOPHY. 307
magnuB a» ^' : 75 x 75 to 1, or as 1778 to 100, nearly.
And the aphelion diftance of this comet from the fun will
he to the mean diftance of the earth from the fun as about 35 to
1; from which data it will be no hard matter to determine the
elliptic orbit of this comet. But thefe things are to be fup-
pofed on condition, that, after the fpace of 75 years, the fame
comet fhall return again in the fame orbit. The other co-*
metsfeem to afcend to greater heights, and to require a longer
time to perform their revolutions.
But, becaufe of the great number of comets, of the
great diftance of their aphelions from the fun, and of the
flown efs of their motions in the aphelions, they will, by their
mutual gravitations, difturb each other; fo that their eccen**
tricities and the times of their revolutions will be fometimes a
little increafed, and fometimes diminifhed. Therefore we
are not to expe<9; that the fame comet will return exadlly in
the fame orbit, and in the fame periodic times : it will be
fufficient if we find the changes no greater than ma}' arife
from the caufes juft fpoken of.
And hence a reafon may be afSgned why comets are not
comprehended within the Umits of a zodiac, as the planets
are ; but, being confined to no bounds, are with various mo-
tions difperfed all over the heavens; namely, to this purpofe^
that in their aphelions; where their motions are exceedingly
flow, receding to greater diflances one from another, they may
fuffer lefs difturbance from their mutual gravitations: and
hence it is that the comets which defcend the loweft, and
therefore move the floweft in their aphelions, ought alfo to
afcend the higheft.
The comet which appeared in the year l680 was in its
perihelion lefs diftant from the fun than by a fixth part of the
fun's diameter ; and becaufe of its extreme velocity in that
proximity to the fun, and fome denfity of the fu^n's atmofphere,
itmufthave fuffered fome refiftance and retardation; and
therefore, being attra<£led fomething nearer to the fun in every
revolution^ will at laft fall down upon the body of the fun.
Nay, in its aphelion,where it moves the floweft, it may fome-
times happen to be yet farther retarded by the attra<^ODs of
X2
308 MATHEMATICAL PRINCIPLES Book l\h
Other comets^ and in confequenceof this retardation defcend
to the fun. So fixed ilars^ that have been gradually wafted
by the liglit and vapours emitted from them for a long time^
; may be recruited by comets that fiUl upon them ; and from
. this frefli fupply of new fuel thofe old ftars, acquiring neW
I fplcndor, may pafs for new ftars. Of this kind are fuch
r fixed ftars as appear on a fuddcn^ and fliine with a wonder-
ful brightnefs at firft, and afleiwards vanifh by little and
little. Such was that ftar which appeared in Cajjliopeia's.
chair ; which Cornelius Gemma.did not fee upon the 8th of
November, 1572, though he was obferving that pari of the
heavens upon that very night, and the fky was perfectly fe<
rene; but the next night (November 9) he faw it fhining
much brighter than any of the fixed ftars, and fcarcely in-
ferior to renns in fplendor. Tycho Brake faw it upon the
llth of the fame month, when it (hone with the greateft
loftre ; and from that time he obferved it to decay by little
and little; and in 1 6 months' time it entirely difappeared*
In the month o{ November, when it firft appeared, its light was
equal to that of Veniis, In the month of December its light
was a little diminiOicd, and was now become equal to that of
Jupiter, In January 1573 it was iefs than Jupiter, and
greater than Siriua; and about the eud of February and the
beginning of March became equal to that ftar. In the
months oi April and May it was equal to a ftar of the fecond
magnitude ; in Juncy July, and Augufi, to a ftar of the third
magnitude ; in September, October, and November, to thofe
of the fourth magnitude ; in December and January 1574 to
thofe of the fifth ; in February to thofe of the (ixth magni-
tude ; and in March it entirely vaniftied. Its colour at the
beginning was clear, bright, and inclining to white; after-
wards it turned a little yellow; and in March 1573 it became
ruddy, like Mars or Aldcbaran : in May it turned to a kind
of dufky whitenefs, like that we obferve in Saturn ; and that
colour it retained ever after, but growing always more and
jnore obfcure. Such alfo was the ftar in the right foot of
SerpentariuSy which Kepler's fcholars firft obferved Sepiem-
ber 30, O.S. l604, with a light exceeding that of Jupiter
\
Book lit. OF NATURAL PHILOSOPHY. 309
though the night before it was not to be feen ; and from that
time it decreafed by little and little, and in 15 or l6 months
entirely difappeared. Such a new ftar appearing with an
unufal l^lenddr is faid to have moved Hipparchus to obferve;
and make a catalogue of, the fixed ftars. As to thofe fixed
ftars that appear and difappearby turns, andincreafeflowlyand
by degrees, and fcarcely ever exceed the ftars of the third mag-
nitude, they feem to be of another kind, which revolve about i
their axes, and, having a light and a dark fide, fhew thofe two *
different fides by turns. The vapours which arife from the
fun, thef fixed ftars, and the tails of the comets, may meet at i
laft with, and fall into, the atmofpheres of the planets by 1
their gravity, and there be condenfed and tunied into water \
and humid fpirits ; and from thence, by a flow heat, pafs t
gradually into the form of falts, and fulphurs, and tindures,^ 1
and mud, and clay, and fand, and ftones, and coral, and other (
terreflirial fubftances.
^ GENERAL SCHOLIUM.
• The hypotbefis of vorti<ies is preffcd with many difficulties.
That every planet by a radius drawn to the fun may defcribe
areas proportional to the times of defcription, the periodic
times of the feveral parts of the vortices fhould obferve the
duplicate proportion of their diftances from the fun; but that
the periodic times of the planets may obtain the fefquiplicate
proportion of their diftances from the fun, the periodic times
of the parts of the vortex ought to be in the fefquiplicate
prbportion of th^ir diftances. That the fmaller vortices may
maintain their leffer revolutions about Saturn, Jupiter, and
other planets, and fwim quietly and undifturbed in the greater
vortex of the fun, the periodic times of the parts of the fun's
vortex Ihould be equal ; but the rotation of the fun and
planets about their axes, which ought to correfpond with the
motions of their vortices, recede far from all thefe proportions.
The motions of the comets are exceedingly regular, are go-
verned by the fame laws with the motions of the planets, and
ican by no means be accounted for by the hypothefis of vor-
tices ; for obmets are c^ried with very eccentrto motions
X 3
«
310 MATHEMATICAL PRlKCIPLSft Book III.
through all parts of the heavens iBdifierently^ with a freedom
that is incooipatible with the notion of a vortex*
Bodies proje6led in our air fuflfer no refiflance but from tlie
air. Witlidraw the air^ as is done in Mr. BoyWs vacnum^
and the refiftance ceafes ; for in this void a bit of fine down
and a piece of folid gold defcend with equal velocity. And
the parity of reafon muft take place in the celeftial fpaces
above the earth^s atmofphere ; in which lpace% where there
is no air to refifl their motions^ all bodies will move with the
greateft freedom ; and the planets and comets will conftantly
purfue their revolutions in orbits given in kind and pofition^
according to the laws above explained; but though thefe
bodies may^ indeed, perfevere in their orbits by the mere laws
of gravity, yet they could by no means have at firfi derived
the regular pofition of the orbits themfelves fix>m thofe laws.
The fix primary planets are revolved about the fun in
circles concentric with the fun, and widi motions dife<^d
towards the fame parts, and almd^ in the &me plane. Ten
moons are revolved about the earth, Jupiter and Saturn^ in
circles concentric with them, with the fame dijne6lion of mo-*
tion, and nearly in the planes of the orbits of thofe planets :
but it is not to be conceived that mere mechanical caufes
could give birth to fo many regular motions, fince the comets
range over all parts of the heavens in very eccentric orbits ;
for by that kind of motion, they pafs eafily through the orbs
of the planets, and with great rapidity ; and in their aphe-
lions, where they move the floweft, and are detained the
longeft, they recede to the greateft diftances from each other,
and thence fuffer the leaft difturbance from their mutual at-
tra6i:ions. This moft beautiful fyftem of the fun, planets, and
comets, could only proceed from the counfel and dominion
of an intelligent and powerful Being. And if the fixed ftars
are the centres of other like fyftems, thefe, being formed by
the like wife counfel, muft be all fubje6l to the dominioB of
One; efpecially fince the light of the fixed fiaurs is of the
fame nature with the light of the fun, and from every {yA&OBi
light pafTes into all the other fyftems: and left the fyftem^
of the fixed ftars fhould, by their gravity^ fall on each other
Book IIL OF NATURAL PHILOSOPHY. 311
mutaally^ be hath placed tfaofe fyftems at immenfe diftances
one from another.
This Being governs all things^ not as the foul of the world,
but as Lord over all ; and on account of his dominion he i$
wont to be called Lord God irUvrqyiqciru^, or Univerfal
Ruhr ; for God is a relative word, and has a refpe<9; to fer-
vants; and Dtity is the dominion of God not over his own
body, as thofe imagine who fancy God to be the foul of the
world, but over fervants. The Supreme God is a Being eter-
nal, infinite, abfolutely perfeA ; but a being, however perfeft,
without dominion, cannot be iaid to be Lord God; for
we fay, my God, your God, th.e God of Ifrael, the God of
Gods, and Lord of Lords ; but we do not fay, my Eternal,
your Eternal, the Eternal of Ifrael, the Eternal of gods; we
do not fay, my Infinite, or my PerfeA : thefe are titles which
have no refpedl to fervants. The word Gocf* ufually figni-
fies Lord; but every lord is not a God. It is the dominion
of a fpiritual being which conftitutes a God : a true, fupreme,
or imaginary dominion makes a true, fupreme, or imaginary
God. And from his true dominion it follows that the true
Grod is a living, intelligent, and powerful Being ; and, from
his other perfe6):ions, that he is fupreme, or moft perfe6J:;
He is eternal and infinite, omnipotent and omnifcient ; that'
is, his duration reaphes from eternity to eternity ; his prefence
from infinity to infinity ; he governs all things, and knows all
things that are or can be done. He is not eternity or infinity,
but eternal and infinite ; he is not duration or fpace, but he en-
dures and is prefent. He endures for ever, and is every
where prefent; and by ^xifi;ing always and every where, he
conftitutes duration and fpace. Since every particle of fpace
is alwaysy and every indivifible moment of duration is every
vffhcre, certainly the Maker and Lord of all things cannot be
iM^
-* Dr. Bfcocl derives the Latin word Deus from the Jrabic du (in the oblique ci^'
4f)% which fi^ni&et Lord* And in this iCtnfe princes ne called gods^ Pfah Izzxii. ver.-
€ t and John %. rtJ. 85. And Mofps h called a ^od to his brother Aaron^ and a god
to Fkofooh QExod. iv. yer. 16 ; and vii. ver. 8). And in the (ame fenfie th^ iouls of
^M priiwes were iotmniyf bj tbt HMthem, oallod godtf but hJUif, beowiliB of th»(r
want of dominion,
X4
312 MATHEMATICAL PEINCIPLES Book III,
*
never and no w/iere. Every foul that has perception is,
though in different times and in different organs of fenfe and
motion, ftill the fame indivifible perfon. There are given
fucceffive parts in duration, co-exiftent parts in fpace, but
neither the one nor the other in the perfon of a man, or his
thinking principle ; and much lefs can they be found in the
thinking fubiian.ce.pf God. Every. man, fo far as he is a
thing that has perception, is one and the fame man during
his whole life, in all and each of his organs of fenfe. God is
the fame God, always and every where. He is omniprefent
not virtually only, but alfo fubjlantially ; for virtue cannot
fubfift without fubftance. In him * are all. things contained
and moved ; yet neither affedls the other : God fuffers
nothing from the motion of bodies ; bodies find no reiiflance
from the omniprefence of God. It is allowed by all that the
Supreme God exifls neceffarily ; and by the fame neceffity he
exifls always and every zihere. Whence alfo he is all iiiniJar,
all eye, all ear, all brain, all arm, all power to perceive, to un-
derfland, and to ad ; but in a manner not at all human, in
a manner not at all corporeal, in a manner utterly unknown
to us. As a blind man has no idea of colours, fo have we no
idea of the manner by which the all-wife God perceives and
underftands all things. He is utterly void of all body and
bodily figure, and can therefore neither be feen, nor heard,
nor touched ; nor ought he to be worQiipped under the repre-
fentation of any corporeal thing. We have ideas of his atr
tributes, but what the real fubflance of any thing is we know
not. In bodies^ we fee only their figures and colours, we
hear only the founds, we touch only their outward fur/aces,
we fmell only the fmells, and tufte the favours ; but their in-
* This was the opinion of the Antients. So Pythagoras, in Ciccr. de Nat. Deot\
lib. i. Thales AnaxagoraSy Virgil, Georg. lib. iv. ver. 220; and .^neid, lib. vi.
ver. 721. Fhilo Ailegor. at the beginning of lib. i. Aratus, in his Phaenom. at
the beginning. So alfa the facred writers ; as St. Paul, Ads, xvii. ver. 27, 28. St.
John\ Gofp. chap. xiv. ver. 2. Mofes, in Deut. iv. ver. 39 ; and x. ver. H.
David, Pfal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job ixii. ver.
12, 13, 14-. Jeremiah xxiii. ver. 23, 24. The Idolaters fuppofed the fun, moon,
sind flats, the fouls of men, and other parts of the world, to be parts of the Supremt
God, &nd therefore to be worfhipped \ but erroneoufljr.
Book III. OF NATURAL PHILOSOPHtY. 313
ward fabftances are not to be known either by our fenfes^ or/
by any reflex a<ft of our minds : much lefs, then^ have we anyi
idea of the fubftance of God. We know him only by his !■
moft wife and excellent contrivances of things^ and final
caufes ; we admire him for his perfecSlions ; but we reverence
and adore him on account of his dominion : for we adore
bim as his fervants ; and a god without dominion^ provi*
dence^ and final caufes^ is nothing elfe but Fate and Nature,
^lind metaphyiical necefiity^ which is certainly the fame al-
ways and every where, could produce no variety of things.
All that diverfity of naturaL things which we find foited to'^
different times and places could arife from nothing but the
ideas and will of a Being neceffarily exifting. But, by way
of allegory, God is faid to fee, to fpeak, to laugh, to love, ta
hate, to defire, to give, to receive, to rejoice, to be angry, to
fight, to frame, to work, to build ; for all our notions of God
are taken frorji the ways of mankind by a certain fimilitude,
which, though not perfcd, has fome likenefs, however. And
thus much concerning God ; to difcourfe of whom from the
appearances of things does certainly belong to Natural I
Philofophy.
Hitherto we have explained the phaenomena of the heavens
and of our fea by the power of gravity, but have not yet .
affigned the caufe of this power. This is certain, that it muft
proceed froi^ a caufe that penetrates to the wery centres of '
the fun and planets, without fuffering the leafl diminution of
its force; that operates not according to the quantity of the
furfaces of the particles upon which it acis (as mechanicai
caufes ufe to do), but according to the quantity of the folid
matter which they contain, and propagates its virtue on all ,
fides to immenfe diftances, decreafing always in the duplicate
proportion of the diftances. Gravitation towards the fun
is made up out of the gravitations towards the feveral parti-
cles of which the body of the fun is compofed ; and in reced-
ing from the fun decreafes accurately in the duplicate pro-
portion of the diliances as far as the orb of Saturn, as evident-
ly appears from the quiefcence of the aphelions of the
planets ; nay, and even to the remotefl aphelions of the co-
*'v^
314 MATHCMAT1CAL PRIlfCIPLB8> &C. Booib IIT.
fluets^ if Ibofe aphelions are alfo qniefcent. Bat hitherto I
hmve not been able lo difcover the canfe of thofe properties
of gravity from phenomena, and I frame no hypothefes;
lor whatever is not deduced from the phaenomena is to be
called an hypotbefis ; and hypothefes, whether metaphyfical
•r phyfical, whether of occult qualities or mechanical^ have
ao place io experimental philofophy. In this philofophy
pariicolar piopofitions are inferred from the phsenomena, and
afterwards rendered general by indu6lion. Thas it was that
the impenetrability, the mobility, and the impnlfive force of
bodies, and the laws of motion and of gravitation, were dif-
eovered. And to us it is enough that gravity does really
exift, and bA according to the laws. which we have explained,
and abundantly ferves to account for all the motions of the
celeftial bodies, and of our fea.
And now we might add fomething concerning a certain
moft fubtle Spirit which pervades and Iks bid in all grofi bo-
dies ; by the force and a<%on of which Spirit the particles of
bodies mutually attraA one another at near diftances, and
cohere, if contiguous ; and electric bodies operate to greater
diftances, as well repelling as attra^ling the neighbouring
corpufcles; and light is emitted, refleAed, refrafted, infletSl-
ed, and heats bodies ; and all fenfation is excited, and the
members of animal bodies move at the command of the will,
namely, by the \nbrations of this Spirit, mutually propagated
along the folid filaments of the nerves, from the outward or-
gans of fenfe to the brain, and from the brain into the muf-
cles. But thefe are things that cannot be explained in few
words, nor are we furniflied with that lufficiency of experi-
ments which is rf^quired to an accurate determination and
demonftration of the laws by which tbis electric and elaftic
Spirit operates.
End of the Mathematical Principles^
APPENDIX,
Among the explications (given by a friend) of fame propo*
fitions in this work not dcmonftrated by the author , the edi-
tor finding the foltowingy has thought it proper to annex
them. Thus, *
To cor. 2, prop. 9h ^ook 1, page IQQ.
JL O find the force whereby a fphere (AdBg) on the diame-
ter AB, attraas the body P. (PI. 19, Fig. 1.)
Let SA = SB = r, PS = d, PE = x, PB = a = d + r,
PA = tf = d — r : therefore a«s = dA — rr ; alfo a + « =:
2d,a — « = 2r; therefore aa — ««=4dr; andSE = d — x,
AE = X — ct, BE = a — x.
Now the force whereby the circle, whofe radius is Ed, at-
PE
trads the body P, is as 1 -^ -^ (by cor. 1, prop, go.).
AndEd*=:(AE x EB = SA* — SE* = rr — dd + 2dx
— XX =) — a« + 2dx — XX. Alfo Pd* = (Ed* + EP*
= 2dx — a« — XX + XX =) 2dx — a«: therefore 7,-f zz
Pd
Therefore 1 —
V^ — a« + 2dx y/ — aat + 2dx
:j or X —
XX
. === is the fluxion of the attradive force of the
%/ — a« + 2dx
•fphere on the body P, or the ordinate of a curve whofe aietk
reprefents that force.
XX
But the fluent of x is x ; and the fluent of ^-^
\/— a« + 2dx
^ a^e -I- dx ^ * * ■ ■ ■ ■ ^ ^ , _
Is 3jj •— aa + 2dx (by Tab. I, Form 4, Caf. «^
Quadr.ofCurv.),
316 APPENDIX.
Ba + dx
Therefore x , , ^/ — a« + lidx is the general ex-
preffion of the area of the curve.
Now let x = a, then area = (a ^"~ ^ — •'^^ + "^^^
Alfo let X = «, then area =: («— "^ , , ^ •— a* + 2d«
d' — r»
=>"3dd"=®-
And the force wliereby the fphere attradb the body P is as
(A — B, or as — =) :_..
3d ' 3PS*
2. The force whereby the fpheroid ADBG attradls the
body P^ may^ in tfie fame mannery be found thus. Let
SC = c.
ITie force of a circle whofe radius is ED to atUaft P is as
PE ^ SC*
1 — -^fz (by cor. 1, prop. 90). Now ED ^^^^^ ^ AEB =:
PD v-j — *• *> i-^-r- ^/- — " — SA'
cc
rr
X — aa + 'idx — XX (by the conies); and PD =i (ER
= EB' + eP = -n^cc + odccx - ccxx _^ ^^ ^^
— atfcc 4- 2dccx + rr — cc X x2 -,, ^ ,,
- — • Therefore (1 —
rr
^=l~ ■ " or) i
PD ♦./ iitfcc 4dcc rr — cc J
y/ ogee
X + XX
rr rr rr
XX
\/ aacc ..idcc rr — cc
^ + X + XX
rr rr rr
is the fluxion of
the attradliye force of the fpheroid on the body P, or the or-
dinate of a curve whofe area is the meafure of that force*
APPENDIX. 317
Now the fluent of x is x ; and (by Caf. 2, Form S, Tab. «,
Quad. Cur.) the fluent of
»{
a/ a^cc 2dcc XT — cc
^ + X +, XX
rr rr jt
8dcc 4dcc 4aacc
__s + _.xv--^v _2dr„+drrxv-a«rrT
4a^gcc rr — cc 4ddc* . a« X cc — rr ^- ddcc
rr rr r
4
— ^ds + dxv — aav ^ 2ds — dxv + aav _,, ^
— . — ) _« m Therefore
— cc — dd + rr cc + dd — rr
x -J Tj is the sjenefal expreflion for the area
cc + da — rr ^ *
of the curve.
^ ^^ ^x. \/ a«cc 2dcc cc — rr
But V = PD = ER = ^ + X 'XX
rr rr • rr
is an ordinate to a conic fe<ftion whofe^'Afcifla is x ; and s,
ffy the areas NMB,NKA, adjacent to the ordinates BM, AK :
put D = s — ff.
Let X = a, or PE = PB = BM ; then v = a, or PD =
■r^T^ -TfcTi/r 11 daa — aaa — 2ds » * -.
PB = BM^ and the area = a + ---tt =A. And
' ' cc 4- dd — rr
let X = a;, or PE = PA = AK ; then v = «, or PD = PA
= AK, and the area =. a A rm = B.
^ cc + dd — rr
And the attra6);ive force of the fpheroid on P is as (A — B
d X aa — ax — aa X a — a — 2d X s — <r
r= a — ce + ■ : — tt
^ cc + dd — rr
2ddr + 2r^ — 2dD ^ 2rcc< + 2d X 2dr — D
zz 2r + \^ ^z) ■
cc + dd — rr ^ ' cc -h dd — rr
But 2d = (a + a =) BM + AK, therefore 2dr =
trapezium ABMK; and D = (s — a* =) area AKRMB;
therefore D — 2dr = mixtilinear area KRMLK = C ;
confequently 2dr — D = — C ; therefore 2d X 2dr ^-^ D
= — 2dC ; therefore the attradlive force of the fpheroid
2rcc — 2dC 2AS x SC* — 2PS X KRMK
on P is as — r-rr = -^ — =i~i — ==- ==i — •
cc+dd — rr §C +P«*— AS
Confequently the attradtive force of the fpheroid upon the
3ia .AWEHmX.
liodyP win be to the mttr^firre fbroe of a fphtnwbofe di*-
rcc— idC I*
aMteriiABiipoDlhefaiiieboJyP>B ^^jj^ . ^ fo ^
AS X gC*^PS X KRMK .^Ag
To 5cAo/. prop. 34^ ftooft ^ j». 84.
For let it be pTopofed to find the vertex of' the cone, a
* firuftum of which has the deferibed property.
Let CF6B be the firoftanij and S the vertex reqwed* (PI.
19, Fig. a.)
Now conceive the medium to copfift of particlei which
firike the fnrface of a body (moving in it) in a direAipn oppo«
fite to that of the motion ; then the refifiande will be the
force which is made Qp of the efficacy of the forces of all the,
ftrokes.
In any line Pp, g|vaUel to the axis of Ae'eone, and rneel*
• Vig its furface in p, take pm of a giveft kngth, for ibe ipace
, deferibed by each point of the cone in a given time: draw,
niq perpendicular to the fide (CF) of the .GOBe> and qa per-
pendicular to pm.
Theiefore the line pm will reprefent the velocity, or force,
with which a particle of the medium ftrikes the furface of the
cone obliquely in p.
But the force mp is equivalent to two forces, the one (mq)
perpendicular, the other (pq) parallel to the fide of the cone ;
which laft is therefore of no effeft.
And the perpendicular force mq is equivalent to two forces,
the one (mn) parallel to the axis of the cone, the other (qn)
perpendicular to it ; which alfo is deftroyed by the contrary
adlion of another particle on the oppofite fide of the cone.
There remains only the force mn, which has any effe6i
in refifting or moving the cone in the direction of its
axis.
Therefore the whole force of a fingle particle, or the effeft
of the perpendicular fl;roke of a particle, upon the bafe of a
circumfcribing cylinder, is to the effed of the oblique ifaroke
APPENDIX. 919
upon the furface of the cone (in p) as mp to rnn, or a^ mp*
to (mp X mn =) mq , or as Cf* to CH •
Now the number of particles ftriking in a parallel direAion
on any furface is as the area of a plane figure perpendicular
to that dire6lion, and that would juft receive thofe ftrokes. .
Therefore the number of particles ftriking againft the
fruftum^ that is^ againft the furfaces defcribed by the rota-
tion of FD and CF, each particle with the forces mp and ma
refpe6lively, is as the circle defcribed by (FD or) OH, and
the annulus defcribed by CH, that is, as OH to CO— OH!
But the whole force of the medium in refifting is the fum
of the forces of the feveral particles.
Therefore the refiftance of the medium, or the whole effi-
cacy of the force of all the ftrbkes againft the end FG of the
fruftum, is to the refiftance againft the convex furface thereof
as (mp X OH* tomn x CO 1^ OH^or as CF V Oh\o Cli
—2 — -=2 —2 CH X CO — oil
X CO — OH, or as) OH to
CF*
Therefore the whole refiftance of the medium againft the
fruftum may be reprefented by (OH* +
CF*
= cF X OH* — CH^ X OH* + CH* x OC* .
CF»
HP X UB* + CH* X P C' ^. , „
' pjjj > which call z ; that is, patting
OC = r, OD = 2a, OS = y, then CH = (2£~S? -)
?^, and OH = 'i-^Zi^^ ^_ r* + r»y*-4ar»yf 4aV
y y f r* + y* *
therefore r* + r*y» — 4ar*y + 4a*r* = r*z + y*z. Cat^
fequently 2r*yy — 4ar*y = 2yzy +y*z + r*z'. But z is a mi-
i
•
320 APPENDIX.
Biinum ; therefore ny ■»• 2arr = zy; confequenily (z =)
r'y — 2ar* r* + r*y* — 4ai*y+4a*r*
'■ y "■ rTTyy .
Hence yy — 2ay = rr ; and, making OQ = QD = a,
tlien (y — a =) QS = ( v/" + aa =) QC.
To the fame Sckol. p. 94.
On the right line BC (PI. 19, Fig 3) fuppofe the parallelo-
grams BGyb, MNvra, of the leaft breadth, to be erecfted,
wbofe heights BG, MN, their diftance Mb, and half the
fum of their bafes ^Mm + ^Bb = a, are given : let half the
difference of the bafes |Mm — |Bb be called x, let G and N
be points in the curve GND; and producing by and my to g
and n (fo that yg =: yn =: b), the points g and n may alfo
be in the fame curve. •
Mow if the figure CDNGB, revolving about the axis BC,
generates a folid, arid that folid moves forwards in a rare and
eUiftic medium from C towards B (the portion of the right
line BC remaining the fame), then will the fum of the refift-
ances againft the furfaces generated by the Itneolss Gg, Nn,
be the leaft poflible, yrhen Gg is to Nn* as BG X Bb to
MN X Mm.
For the force of a particle on Gg and Nn to move them in
1 I
the diredlion BC is as — -^ and ==;", and the number ofpar-
Og Nn
^cles that ftrike in the fame time on the furfaces generated
by Gg and Nn are as (the annuli defcribed by gy and ny, that
is, as BG X gy and MN X ny, or as) BG and MN; there-
fore the refiftances affainfl thofe furfaces are as t=i — to — »
"" Gg* Nu^
. . r 77- . r ^^^ ^G MN
that is (puttmg y for ug% and z for JNn Jj as to —y- •
^ . n . . . nn BG MN^ .
But the fum of thefe refinances ( 1- -^77^7 is a mini-
mum. Therefore — BG x ^— MN x — = 0, or MN
yy zz
z
><;; = — EG X ;£. Buty = (Gg* = Bb' + yg» =) aa
zz yy
yy.r/,-xix.K./ii
1 ■
I
I ■
>i
APPENDIX. 321
— 2ax + XX + bb; and z = (No* = Mm* + vn* = )
- • • • •
aa + Sax + xx -f bb ; therefore y = 2xx — Sax, and z
, MN .
=: 2ax + 2xx : confeqaently — — x 2x x a + x =
BG ^. ,MN ^ MN ^,
— X fix X a — X ; or ( xa + x = ) x Mm
yy - - 2z 22
Tir^ -^— ^— BG _
= (— X a — X =) — X Bb. Tbeiefore (yy) Gg* : (zz)
K^* : : BG X Bb : MN X 3^Im.
ConieqiieDtiy, that the fom of the refifiances againfi the
farfiuxs generated by the lineols G^ and Nn maybe the
leafi poffible, Gg^ moftbe to Nd^ as GBb to NMm.
Wherefore^ if 7g be made equal to yG, fo that the angle
yGg may be 45^ and die angle BGg 135**; alfo Gg* =
fiyg*, and Gg* = ^*7g* ; *hen 47g : Nn* : : GBb : N>Im ;
and fince GR is parallei to Nn, and BG, BR parallel to ny,
Ny ; alfo nv = gy = 7G ; it follows that (ny = 9^6 =:) Bb :
(N» = ) Mm : : BG : BR, thaefoie Bb = °^ ^^^ —i
alfo (nv = ) 7G : Nn : : BG : GR. Confeqnently
^n GR* SSlm MN x BR
X BR is to GR' as GR to MN.
EXD OF TOL. II.
Vot. n.
INDEX TO VOL. II.
Page
^Equinoxes, their pr^eceffion.
the caufe of chat motion (hewn ••••••.••••• 197
the quantity of that motion computed firom the caoTed. • • . 252
AiR> its denfity at any height^ colledted by prop. 22, book 2, and
its denfity at the height of one femi-diameter of the earth,
(hewn 290
ks eladic force, what caufe it may be attributed to 60
its gravity compared with that of water 290
' its refiftancey collected by experiments of pendulums. « 77
the fame more accurately by experiments of falling bodies^ and
a theory , 128
Attraction of all bodies demonftrated 176
the certainty of this demondration (hewn I60
che caufe or manner thereof no where defined by the
author.... .,. 313
the common centre of gravity of the; Qarth^ fun, and all the pla-
nets, is atTeft, confirmed by cotw 2, prop. 14, book 3 182
the common centre of gravity of th^ eajth and moon goes round
the orlh magnus •••.•• 183
Its diftance from the earth and from the moion • • • . . 245
Comets;, a fort of planets, not meteors .261, 285
higher than the moon, andin theplanetary regions. ••.•• •• 254
their didance how coUedled very nearly by obfervations. • . . • 255
more of them obferved in the hemifphere towards the fna
than in the oppofite hemifphere ; and how this comes to
pafs... i«. 260
(hine by the fun's light reflected fcom.them. • ibid
furrounded with vaft atmofpheres. . . • 258, 261
thofe which come neareft to the fun probably the leafl 297
why they are not comprehended within a zodiac, like the
planets, but move differently into all parts, of the
heavens. • 307
may fometimes fall into the fun, and afford a new fupply of
fire ibid.
the ufe of them hinted 294
Y2
»DBZ*
CoMtTS more io conie iedioiMf hftTtag^ dieir fiici io the fan's cen-
tre, and by radii drawn to the fun defcribe areas proportional
to the times. Move in ellipses if they come round again in
their orbits, but thefe ellipfiss will be near to parabolas 262
CoM£T*s parabolic trajedory found from three obfervations
given 268
corre£^cd when found 298
place in a ptrabola found to agiven time 262
velocity compared yrith the velodcy of the planets. ibid
CoMETs'Taits direded firom the fon 289
brigbteil and Urged immediately after their pai£^ through
the neighbourhood of the fan 286
their wonderful rarity • 291
their origin and nature 258
in what fpaoe of time they aicend from the heads..... 291
CoMtT of the years 1664 and l665
the obfervations of its motion compared with the theory 299
of ibe years' 1 680 and l68i
obfervadons of its motion 27 1
its modoa computed in a parabolio orfatt • •••.•- 276
in an elliptic orbit ; 278
its trajedory, and its tail in the fef«dl parts of its oibit, deli-
neated 2S3
of the year l682
its motion compared with the theory 505
feems to have appeared in the year I6079 andlilely to retom
again after a period of 75 years 306, 307
of the year l683
its motioa compared with the theory. 304>
of the year 1723
its motion compared with the theory. .., , 306
CoavATVRc of figures how eftimated •.25> 209
Descent of heavy bodies in vacuo, bow much it is 187
andafcer.t of bodies in refifling mediums. . .2, 17> 18, 37» 39> 1 15
Earth, its dimenijon by N&rivnoJf by Pkart, and by Cqfiti,, . . 187
its figure difcovered, with the proportion of its diameters, and
the meafure of the degrees upon the median .1 87» 191
INDEX. .
Page ,
£ AftTH, the excefs of its height at the equator aboife its height at
the poksi. ...;..; ......•.;..:..'. 190, ^^
its greateA^ and leail remi<(diameter I90
Its-mean (emivdiaraeter. ibid
the globe of earth more denfe t han if ir t^as ehtirdy water ... 180
the nutation of its axis. • • , • I97
the annual motion thereof in the orbis magnus demonflrated. . • 301
the eccentricity thereof how much 245'
the melion of its aphelion how much 186
FtuiDSj-the laws of their denfity and comprelHon Aiewn . . Sed. V, 50
their motion in running out at ^ hole in a veiTel determined • . • Q%
Force, centrifugal force of bodies oti the earth's equator, how
greatk 187
God, his nature 31£
Gravity ofa different nature from magnetical fdrce '• 176
the caufe of it not afligned «....«• 313
tends towards all the planets '. . 1 171
from the furfaces of the planets upwards decreaies in the dupli-
cate ratia of the diftances firom the centre ...» ISO
from 'the &me downwards decreaies dearly in the ilmple ratio
of the fame '.' . ibid
tends towards all bodies^ and is proportional to the quantity of
matter in each ^ I76 '
is the force by which the moon is retained in its orbit 169
<he fame proved by in accurate calculus ■. 245, S^t)
is the force by which the primary planets atid theTatellites of
- 'Jupiter and Saturn are retained in theit oVbit§. . '. . • . * '. ^. . 171
IttATf an iron rod increafes in length by heat. ... ^ ...'.'...•'.'. . l^ '
^ of the fun, how great at different diftaildeSs (rdin the fun ...••'. '285
how great in Mercury ; ; • . • i 1 « ,', . 180
how- great in the comet of I68O, whfefi in its perihelbn . . ... 285
I^iAVSN s are void of anyfenfible refiftance, 181, 261, 293 ; and,
therefore, of alraoft any corporeal fluid whatever. ... 128, 129
fuffer light to pais through them without any refradion 286 ^
HToaoflTiLTics, the principles thereof delivered SeGL V. 50
Htpothisis of what kind foever rejeded from this philofophy.-. 314
Jupiter^ its periodic time; ; . . , 165
its diftance from the fun 166'
mots*
Page
Jcf r iriR, ks apptfcot dnwctcr *•• 163
iu true diameter* •• » 179
its attnAive force, how gratt 17S
the wcighu of bodies 00 itsfur£ioe ••.... jjg
its denHty , ibid
its quantity of matter. • ••••••...• ibid
its pcrturbatioo by Saturo, how much « ••••• 183
the proponion of its diameters exhibited fay computatioo . • • • . 1 9 1
and compared with oUcrvatioos « ibid
its routioo about itt aus, in what time performed .•....•.. ibid
the caufe of its belts hinted at 26l
Light, doc caufed by the agiution of any ethereal medium. . • . 144
Magketic force 69, 176, 247
Mars, its periodic time 1^5
its diflance from the fun 166
the motion of its aphelion ••.•• ISS
MATTEa» its extcnfion, hardnefs, impenetrabtlityy mc^nlity^ twr
mtrile, gravity, how diicovered 1^1
fubtle matter of Defcarte^ enquired into 84
Meicubt, its periodic time l65
its diftance from the fmi • 166
the motion of its aphelion • t... 186
Method of fiuxiocs 13
diiTerential 263
of finding the quadratures of all curves very nearly true 264
of converging feries applied to the (bhition of di£cidt
problems 25, 225
Moon, the figure of its body cc^e^ed by calculation 247
its Kbrations explained 1$6
its mean apparent diameter •• 245
its true diameter ibid
weif^ht of bodies on its furface ibid
its denliry ibid
its quantity of matter ilnd
its mean diftance from the earth, how many greateft (emi-
diameters of the earth contained therein ibid
how many mean iemi-diaraeters ••••••• 247
ks force to move the fea how great ....••••••• •• 241
IMDftX.
Page
Moon not perceptible in experiments of pendulumsi or any flatical or
hydroftatical obfervations • • • . 244^
Its periodic time •. ^ 24&
the time of its fynodical revolution 208
its motioQSy and the inequalities of the fame derived from their
caufes 197, 234
revolves more flowly, in a dilated orbit, iphen the earth is in
its perihelion ; and more fwiftly in the aphelion the &me,
its orbitheing contracted. 197> 234, 235
revolves more flowly in a dilated orbit when the apogseoii is in
the fyzygies with the fun ; and more fwiftly in a contraded
orbit when the apogaeon is in the quadratures • • 236
revolves more (lowly, in a dilated orbit, when the node is in
the fyzygies with the fun ; and more fwiftly, in a cotitra6l-
ed orbit, when the node is in the quadratures 237
moves flower in its quadratures with the fun, fwtfter in the fy-
zygies ; and by a radius drawn to the earth defcribes an
area, in the firfl cafe lefs in proportion to the time, in the laft
cafe greater .^ I97
the inequality of thofe areas computed » . • • . 266
its orbit is more curve, and goes farther from the earth in the
firil cafe ; in the lad cafe its orbit is lefs curve, and comes
nearer to the earth ^ * I97
the figure of this orbit, and the proportion of its diameters
colledted by computation , , 209
a method of finding the moon's diftance firom the earth by
its horary motion ibid
its apogaeon moves more flowly when the earth is in its aphe-
Uon, more fwiftly in the perihelion I98, 235
its apogaeon goes forward mod fwiftly when in the fyzygies
with the fun ; and goes backward in the quadratures. . 198, 237
its eccentricity greatell when the apogseon is in the fyzygies
with the fun ; lead when the fame is in the quadratures, 1 98, 237
its nodes move more flowly when the earth is in its aphelion,
and more fwiftly in the perihelion 198, 235
its nodes are at reft in their fyzygies with the fun, and go back
moft fwiftly in the quadratures I98
the motions of the nodes and the inequalities of its motions
(computed from the theory of gravity 214, 218, 222, 22f
2
INDXX.
Page
MooN> tbe fame from a di^Terent principle 226
the variations of the inclination computed from the theory of
gravity , 230, 232
the equations of the moon's motions for aftronoroical ofes • • • 235
the annual equation of the moon's mean motion ibid
the firft femi-annual equation of the lame . , • • • -• 236
the fecond\emi<annual equation of the fame 237
the firfl equation of the moon's centre 23S
the fecond equation of the moon's centre 239
Moon's firft variation « 212
the annual equation of the mean motion of its apogee 235
the feroiannual equation of the fame. •'•••.•< 237
the femi-annual equation of its eccentricity ibid
the annual equation of the mean motion of its nodes • » 235
the femi-annual equation of the fame 226
the femi-annual equation of the indinauon pf the orbit to the
ecliptic . • 7 • « . 234
the method of fixing the theory of the lunar motions from ob-
fervations » •••• 259
Motion of bodies refifled in the ratio of thevelocides. . • Se^. I. 1
in the duplicate ratio of the velocity Se^* IL 9
partly in the iimple and partly in the duplicate ratio of the
fame Sed. III. 34
of bodies proceeding by their vis injita alone in refifting me-
diums 1, 2, 9, 11, 34, 35, 9^
of bodies afcending or defcending in right lines in refifling me*
diums, and afted on by an uniform force of gravity, 2, 17, 18, 37> ^9
of bodies projected in refiding mediums, and adled on by an uni-
form force of gravity 5, 21
of bodies revolving in refifting mediums Se6^. IV. 43
of funependulous bodies in refifling mediums Se6^. VI. 62
and refiftance of fluids Sea, VII. 87
propagated through fluids Seft. VIII. 130
of fluids after the manner of a vortex, or circular. .Se€l. IX. 146
Pendulums, their properties explained Sedl. VI. 62
the diverfe lengths of ifochronous pendulums in different lati-
tudes compared among themfelves, both by obfervations
and by the theory of gravity 192 to 197
INDEX.
Pag©
Planets not earned about by corporeal TortiCes. • • 157
Planets, Primary^ furround the fun • l65
move in ellipfes whofe focus is in the fun's centre • 1 83
by radii drawn to the fun defciibe areas proportional to the
times l66y 185
revolve in periodic times that are in the fefquiplicate propor*
tion of the diftances from the fun « . « ^ . l6S
are retained in their orbits by a force of gravity which refpedts
the fun, and is reciprocally as the fquare of the diflance from
the fun's centre . . . , « « « • • l67> 171
PLAN£TSy SECONDARY) movc in cUipfes having their focus in the
centre of the piimary • ig/
by radii drawn to their primary defcribe areas proportional to
the times , . • . . ^,. . . l62, l64; l67
revolve in periodic times that are in the fefquiplkate prc^por-
tion of their diftances frqm the primary « • l62> 1^4
Planets, their periodic times ..•>••..••••«,.,«.«.•• ^ • l6S
their dift^nc^s from the fun «•««••«•««.«•,•• c. l66
the i^phelia.and nodes of th^ orbits do almoft reft • « • • i^ • * . • 185
their orbits determined* ...•••..#,.,.,,,••••«••, « 186
Uve way o£ finding their pls^^es in their orbits. •••«••• 1 17 to 122
. their ^enfity fuited to the heat they re.ceive from the fun 1 80
their diurnal revolutions equable 186
their axes lefs than the diameters that fiand upon them at right
angles; .•*....«• 187
Projectiles move in parabolas when the reiiftance of the me-
dium is taken away. ^ . . • . ^ 27
their n^Qtiotis in refilling mediums. 5, 21
Pulses of the air, by which founds are propagated, their intervals
Qr breadths determined ^ ..... ^ ^ ......... 144, 146
thefe intervals in (bunds made by open pipes probably equal ta
twice the length of the pipes ibid
Qualities of bodies how difcovered, and when to be fuppofed
univerfal l6^
Resistance, the quantity thereof in mediums not continued. ... . gs
in continued mediums , 192
in mediums of any kind whatever 97
Resistances, the theory thereof confirmed by experiments of
pendulums , 74 to 85
by experimenu of fidling bodies. •..•••• 115 to 129
Vol. IL Z
INDEX.
Page
RisiSTANCB of mediums is as their denfltyy e£t^ru fart'
^ ^h 85, 89, 95, 113, 128
is in the duplicate proportion of the velocity of the bodies re-
fifted, extern farihu f), 75y 8p, 05, 113, 122
b in the duplicate proportion of the diameters of fpherical bo-
dies refilled, ceteris paribus 79, 81, 95, 1 13
of fluids threefold, arifes either from the ioa^ivity of the tiuid
matter, or the tenacity of its parts, or fri^ion . . . .- 42f
the refinance found in fluids, dmoft all of the firft kind, 85, 127
cannot be diminiflied by the fubtilty of the parts of the fiuid,
if the denfity remain ...••••• 128
of a g^lobe, what proportion it bean to thai of a cylinder, in
mediums not continued. • • 92
in cooiprefled mediums 112
•fa globe in mediums not continued 95
in comprefled mediums. •••••• 113
how found by experinients* • • ••••»••• 115 to Ii$8
to a fruftum of a cone, how made the leaft pcffihie... 94
what kind of (oM it is that tteeis with theleaft*. ^5
RuLBs of philofophy v > v <.•.... • A\ .'.'......• I60
Satellites, the greateft beliOcentrib tekngadon ^ Ji^ker's
fatcUites l63
the greated heliocentric elongation of the ^tr^^/w fatellite
firom Saturn's centre .- 178
the periodic times of Jupiter's fatellites, and their diflances
from his centre l62, 1^3
the periodic times of Saturn's (atellites, and their diilances
from his centre * 1(>4>, 1^5
the inequalities of the motions of the fatellites of Jupiter and
Saturn derived from the motions of the moon 197
Saturn, its periodic time l65
its diflance from tlie fun 1^6
its apparent diameter l65
its true diameter 179
its attradiive force how great • 17^
the weight of bodies on its furface • 179
its denfity , • ibid
its quantity of matter ibid
its perturbation by the approach of Jupiter how great 184
tbcjapparem diameter of its ring. • \6S
IHDfiX.
Page
Shadow of the earth to be augmented in lunar ecltpfes, becanfe of
the refra<flioD of the atmofphere ..••• 239
Sun, moves round the common centre of gravity of ail the planets. 182
the periodic time of its revoludan about its axis. • . • • 1 86
its mean apparent diameter. • ••«••.. ^. • 245
its true diameter • . . . i78
its horizontal parallax »••.•'• ibid
lias a mendrual parallax • • 184
its attradtive force how great • • 173
the weight of bodies on its furface • 179
its denlity • • ibid
its quantity of matter • • . • ibid
its force to difturb the nootions of the moon. • l69» 205
its force to move the Tea. •■.;•• • 240
Sounds, their nature explained* • 133, 136, 140^ 142,145^ 144, 145
not propagated in direSum 152
cauled by the agitation of the air 144
their velocity computed • 144, 145
ibmewhat fwifter by the theory in fummer than in winter. ... 146
ceafe inmiediately, when the motion of the fonorous body
ceafes •..««••• 140
how augmented in fpeaking trumpets 14S
Space, not equally full 175
Spiral cutting all its radii in a given angle, by what law of centri-
petal force, tending to the centre thereof, it may be defcrib-
ed by a revolving body « 43, 48
Spirit pervading ail bodies, and concealed within them, hinted at,
as required to folve a great many phenomena of Nature. . . 314
Stars, the fixed (lars demonflrated to be at reil | 185
their twinlcling what to be afcribed to. . • • » 287"
new ftars, wlience they may arife 307
Substances of all things unlcnown » 313
Tides of the fea derived from their caufe 199, 240, 241
A Vacuum proved, or that all fpaces (if faid to be full) are not
equally full , ., I75
Velocitt, the greatefl that a globe falling in a refiftiog medium
can acquire ; •• •• • . . . ,113
Venus, its periodic time ^ 1^5
its diftance from the fun \(^q
the motion of its aphelion^ 13^
2
tllDIX.
Page
VoiTicct,tKeir ttturc and oonfimKiM eKamined 309
Watis, the velocity with which they afeprofN^^uedtMi the fiiper-
fides of Ragoaiit water 1S6
WftiCHTS ofhodiet towaidi the fin, the earthy or any planet, aie,
at equal difiancat fiem the centre, at the quantities of mat-
ter in the bodies «•• 172
they do not depend upon the forms and teztnres of bodies. «. . 173
of bodies ia difiereat regions of the eanh found out, and com-
pand together , • , •• ; . « • 192
ERRATUM.
Page 120, line 3 from bottom, fiMT ' parts, read l^partsu
KAizbt & Coinpcois Middle Strec%
Ctotii Fait.
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