mediately drawn back, and converted again towards the earth.
BOOK III] OF NATURAL PHILOSOPHY.
LEMMA I.
If APEp represent the earth uniformly dense, marked with the centre C,
the poles P, p, and the equator AE; and if about the centre C, with
the radius CP, we suppose the sphere Pape to be described, and Q li to
denote tJie plane on which a right line, drawn from the centre of the
sun to the centre of the earth, i/isists at right angles ; and further
suppose that the several particles of the whole exterior earth PapAP^pE,
without the height of the said sphere, endeavour to recede towards t/iis
side and that side from the plane Q.R, every particle by a force pro
portional to its distancefrom that plane ; I say, in the first place, that
the whole force and efficacy of all the particles that are situate in AE,
the circle of the equator, and disposed uniformly without the globe,
encompassing the same after the manner of a ring, to iclieel 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 greatest distance from
the plane Q,R, to wheel the earth about its centre with a like circular
motion, as I to 2. And that circular motion will be performed about
an axis lying in the common section of the equator and the plane Q,R.
For let there be described from the centre K, with the diameter IL, the
semi-circle INL. Suppose the semi-circumference INL to be divided
into innumerable equal parts, and from the several parts N to the diameter
Q
I K ML,
IL let fall the sines NM. Then the sums of the squares of all the sinea
NM will be equal to the sums of the squares of the sines KM, and both
sums together will be equal to the sums of the squares of as many semidiameters
KN ; and therefore the sum of the squares of all the sines NM
will be but half so great as the sum of the squares of as many semi-diam
eters KN.
Suppose now the circumference of the circle AE to be divided into the
like number of little equal parts, and from every such part P a perpen
dicular FG to be let fall upon the plane QK, as well as the perpendicular AH from the point A. Then the force by which the particle F recede*
456 THE MATHEMATICAL PRINCIPLES [BOOK IIL
from the plane QR will (by supposition) be as that perpendicular FG ; and
this force multiplied by the distance CG will represent the power of the
particle F to turn the earth round its centre. 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 GO to AH X HC ; that is, as FC 2 to AC2
: and therefore
the whole power of all the particles F, in their proper places F, will be to
the power of the like number of particles in the place A as the sum of all
the FC 2 to the sum of all the AC2
, that is (by what we have demonstrated
before), as 1 to 2. Q.E.D.
And because the action of those particles is exerted in the direction of
lines perpendicularly receding from the plane QR, and that equally from
each side 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 II.
The same things still supposed, I say, in the second place, that the total
force or poiver of all the particles situated every where about the sphere
to turn the earth about the said axis is to the whole force of the like
number ofparticles, uniformly disposed round the whole circumference,
of the equator AE in the fashion of a ring, to turn the whole earth
about with the like circular motion, as 2 to 5.
For let IK be any lesser circle parallel to
the equator AE, and let L/ be any two equal
particles in this circle, situated without the
sphere Pape ; and if upon the plane QR,
which is at right angles with a radius drawn
to the sun. we let fall the perpendiculars LM,
Im, the total forces by which these particles
recede from the plane QR will be propor
tional to the perpendiculars LM, Im. Let
the right line LZ be drawn parallel to the
plane Papc, and bisect the same in X ; and
through the point X draw Nw parallel to the plane QR, and meeting the
perpendiculars LM, Im, in N and n and upon the plane QR let fall the
perpendicular XY. And the contrary forces of the particles L and I to
wheel about the earth contrariwise are as LM X MC, and Im X mC ; that
is, as LN X MC + NM X MC, and In X mC nm X mG or LN X
MC + NM X MC, and LN x mC NM X mC, and LN X Mm
NM X MC" 4- raC, 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 MA/?,, or 2LN X NX7 is to 2AH X HC, the force of two particles
of the same size situated in A, as LX2 to AC 2
; and the negative part NM
BOOK 111. OP NATURAL PHILOSOPHY. 45?
X MC T wC^or 2XY X CY, is to 2AH X HC, the force of the same
two particles situated in A, as CX 2 to AC 2
. And therefore the difference
of the parts, that is, the force of the two particles L and /, taken together,
to wheel the earth about, is to the force of two particles, equal to the
former and situated in the place A, to turn in like manner the earth round,
as LX2 CX2 to AC 2
. But if the circumference IK of the circle IK
is supposed to be divided into an infinite number of little equal parts L,
all the LX2 will be to the like number of IX 2 as 1 to 2 (by Lem. 1) ; and
to the same number of AC 2 as IX 2 to 2AC2
; and the same number ol
CX2 to as many AC2 as 2CX 2 to 2AC 2
. Wherefore the united forcet
of all the particles in the circumference of the circle IK are to the joint
forces of as many particles in the place A as IX 2 2CX2 to 2AC 2
; and
therefore (by Lem. 1) to the united forces of as many particles in the cir
cumference of the circle AE as IX 2 2CX 2 to AC 2
.
Now if Pp. the diameter of the sphere, is conceived to be divided into
an infinite number of equal parts, upon which a like number of circles
IK are supposed to insist, the matter in the circumference of every circle
K will be as IX2
; and therefore the force of that matter to turn the
earth about will be as IX 2 into IX 2 2CX2
: and the force of the same
matter, if it was situated in the circumference of the circle AE, would be as
IX 2 into AC 2
. And therefore the force of all the particles of the whole
matter situated without the sphere in the circumferences of all the circle?
is to the force of the like number of particles situated in the circumfer
ence of the greatest circle AE as all the IX 2 into IX2 2CX 2 to as
many IX 2 into AC2
;
that is, as all the AC 2 CX2 into AC2 3CX 2
to as many AC2 CX2 into AC2
: that is, as all the AC 4 4AC 2 x
CX2 + 3CX4 to as many AC 4 AC 2 X CX2
;
that is, as the whole
fluent quantity, whose fluxion is AC 4 4AC 2 X CX 3 + 3CX 4
, to the
whole fluent quantity, whose fluxion is AC 4 AC 2 X CX 2
; and, there
fore, by the method of fluxions, as AC 4 X CX fAC 2 X CX 3 +
|CX. 5 to AC 4 X CX i-AC
2 X CX3
;
that is, if for CX we write the
whole Cp, or AC, as T
4jAC 5 to fAC 5
;
that is, as 2 to 5. Q.E.D.
LEMMA III.
The same things still supposed, I say, in the third place, that the mo
tion of the i^hole earth about the axis above-named arising from the
motions of all the particles, will be to the motion of the aforesaid ring
about the same axis in a, proportion compounded of the proportion of
the matter in the earth to the matter in the ring ; and the proportion
of three squares of the quadrantal arc of any circle to two squares
of its diameter, that is, in the proportion of the matter to the matter,
and of ttie number 925275 to the number 1000000.
the motion of a cylinder revolved about its quiescent axis is to the
*68 THE MATHEMATICAL PRINCIPLES [BOOK III.
motion of the inscribed sphere revolved together with it as any four equal
squares to three circles inscribed in three of those squares ; and the mo
tion of this cylinder is to the motion of an exceedingly thin ring sur
rounding both sphere and cylinder in their common contact as double the
matter in the cylinder to triple the matter in the rir^j ; and this motion
of the ring, uniformly continued about the axis of the cylinder, is to the
uniform motion of the same about its own diameter performed in the
same periodic time as the circumference of a circle to double its diameter.
HYPOTHESIS II.
If the other parts of the earth were taken away, and the remaining ring
was carried alone about the sun in, the orbit of the earth by the annual
motion, while by the diurnal motion it ivas in the mean time revolved
about its own axis inclined to the plane of t/te ecliptic by an angle
of 23i decrees, the motion of the equinoctial points would be the
same, whether the ring were fluid, or whether it consisted of a hard
and rigid matter.
PROPOSITION XXXIX. PROBLEM XX.
ToJind the precession 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" 35 " 16iv
. 36V
.
; the half of
which, 8" 17 " 38 v
. 18V
. (for the reasons above explained) is the mean ho
rary motion of the nodes in such an orbit, which motion in a whole side
real year becomes 20 11 46". Because, therefore, the nodes of the moon
in such an orbit would be yearly transferred 20 11 46" in antecederttia ;
and, if there were more moons, the motion of the nodes of every one (by
Cor. 16, Pro]). LXVI. Book 1) would be as its periodic time; if upon the
surface of the earth a moon was revolved in the time of a sidereal day,
the annual motion of the nodes of this moon would be to 20 31 46" as
23h
. 56 , the sidereal day, to 27 !
. 7h
. 43 , the periodic time of our moon,
that is, as 1436 to 39343. And the same thing would happen to the
nodes of a ring of moons encompassing the earth, whether these moons
did not mutually touch each the other, or whether they were molten, and
formed into a continued ring, or whether that ring should become rigid
and inflexible.
Let us, then, suppose that this ring is in quantity of matter equal to
the whole exterior earth PctpAPepR, which lies without the sphere Pape
(see fig. Lem. II) ; and because this sphere is to that exterior earth as Cto
AC 2 aC2
, that is (seeing PC or C the lea^t semi-diameter of the
earth is to AC the greatest semi-diameter of the same as 229 to 230), as
52441 to 459 : if this ring encompassed the earth round the equator, and
both together were revolved about the diameter of the ring, the motion of
HOOK III.] OF NATURAL PHILOSOPHY. 459
the ring (by Lcm. Ill) would be to the motion of the inner sphere as 459
to 52441 and 1000000 to 925275 conjunct!}, that is, as 4590 to 485223;
and therefore the motion of the ring would be to the sum of the motions
of both ring and sphere as 4590 to 489813. Wherefore if the ring ad
heres to the sphere, and communicates its motion to the sphere, by which
its nodes or equinoctial 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 diminished in the same propor
tion. Wherefore the annual motion of the equinoctial points of the body,
composed of both ring and sphere, will be to the motion 20 11 46" as
1436 to 39343 and 4590 to 489813 conjunctly, 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 equinoctial points of the ring recede
(that is, the forces SIT, in fig. Prop. XXX), are in the several particles
as the distances of those particles from the plane Q,R ; and by these forces
the particles recede from that plane : and therefore (by Lem. II) if the
matter of the ring was spread all over the surface of the sphere, after the
fashion of the figure PupAPepl^, in order to make up that exterior part
of the earth, the total force or power of all the particles to wheel about
the earth round any diameter of the equator, and therefore to move the
equinoctial points, would become less than before in the proportion of 2 to
5. Wherefore the annual regress of the equinoxes now would be to 20
11 46" as 10 to 73092 ; that is. would be 9" 56 " 50iv
.
But because the plane of the equator is inclined to that of the ecliptic,
this motion is to be diminished in the proportion of the sine 91706
(which is the co-sine of 23 1 deg.) to the radius 100000 ; and the remain
ing motion will now be 9" 7 " 20iv
. which is the annual precession of the
equinoxes arising from the force of the sun.
But the force of the moon to move the sea was to the force of the sun
nearly as 4,4815 to 1
; and the force of the moon to move the equinoxes
is to that of the sun in the same proportion. Whenoe the annual precession
of the equinoxes proceeding from the force of the moon comes out 40"
52" 521V
. and the total annual precession arising from the united forces
of both will be 50" 00" 12iv
. the quantity of which motion agrees with
the phaenomena ; for the precession of the equinoxes, by astronomical ob
servations, is about 50" yearly.
If the height of the earth at the equator exceeds its height at the
poles by more than 17| miles, the matter thereof will be more rare near
the surface than at the centre
; and the precession of the equinoxes will
be augmented by the excess of height, and diminished by the greater rarity,
And now we have described the system of the sun, the earth, moon,
and planets, it remains that we add something about the comets.
460 THE MATHEMATICAL PRINCIPLES [BOOK IIL
LEMMA IV
That the comets are higher tliau tJie moon, and in the regions of the
planets.
As the comets were placed by astronomers above the moon, because they
were found to have no diurnal parallax, so their annual parallax is a con
vincing proof of their descending into the regions of the planets ; for all
the comets which move in a direct course according to the order of the
signs, about the end of their appearance become more than ordinarily slow
or retrograde, if the earth is between them and the sun ; and more than
ordinarily swift, if the earth is approaching to a heliocentric opposition
with them ; whereas, on the other hand, those which move against the or
der of the signs, towards the end of their appearance appear swifter than
they ought to be, if the earth is between them and the sun ; and slower,
and perhaps retrograde, if the earth is in the other side of its orbit. And
these appearances proceed chiefly from the diverse situations which the
earth acquires in the course of its motion, after the same manner as it hap
pens to the planets, which appear sometimes retrograde, sometimes more
slowly, and sometimes more swiftly, progressive, according as the motion of
the earth falls in with that of the planet, or is directed the contrary wav.
If the earth move the same way with the comet, but, by an angular motion
about the sun, so much swifter that right lines drawn from the earth to
the comet converge towards the parts beyond the comet, the comet seen
from the earth, because of its slower motion, will appear retrograde ; and
even if the earth is slower than the comet, the motion of the earth being
subducted, the motion of the comet will at least appear retarded
; but if the
earth tends the contrary way to that of the cornet, the motion of the comet
will from thence appear accelerated; and from this apparent acceleration,
or retardation, or regressive motion, the distance of the comet may be in-
F c B A ferred in this manner. Let TQA,
TQ,B, TQ,C, be three observed lon
gitudes of the comet about the time
of its first appearing, and TQ,F its
last observed longitude before its
disappearing. Draw the right line
ABC, whose parts AB, BC, intercepted
between the right lines QA
and Q.B, QB and Q.C, may be one to the other as the two times between
the three first observations. Produce AC to G, so as AG may be to AB
as the time between the first and last observation to the time between the
first and second ; and join Q.G. Now if the comet did move uniformly in
a right line, and the earth either stood still, or was likewise carried foru
ards in a right line by an uniform motion, the angle TQG would be tht
BOOK 111.] OF NATURAL PHILOSOPHY. 401
longitude of the comet at the time of the last observation. The angle,
therefore, FQG, which is the difference of the longitude, proceeds from the
inequality of the motions of the comet and the earth ; and this angle, if
the earth and cornet move contrary ways, is added to the angle TQ,G, and
accelerates the apparent motion of the comet ; but if the comet move the
same way with the earth, it is subtracted, and either retards the motion ol
the comet, or perhaps renders it retrograde, as we have but now explained.
This angle, therefore, proceeding chiefly from the motion of the earth, is
justly to be esteemed the parallax of the comet; neglecting, to wit, some
little increment or decrement that may arise from the unequal motion of
the comet in its orbit : and from this parallax we thus deduce the distance
of the comet. Let S represent the sun, acT v
the orbis tnagnus, a the earth s place in the
first observatiun, c the place of the earth in
the third observation, T the place of the
earth in the last observation, 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 longitude of the comet
at the time when the earth is in T ; join ac,
and produce it to g
1
, so as ag may be to ac
as AG to AC ; and g will be the place at
which the earth would have arrived in the
time of the last observation, if it had con
tinued to move uniformly in the right line
ac. Wherefore, if we draw g T parallel to TT, and make the angle T^V
equal to the angle TQ,G, this angle Tg\ will be equal to the longitude of
the comet seen from the place g, and the angle TVg- will be the parallax
which arises from the earth s being transferred 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 same thing may be deduced from the incurvation of the way of the
comets ; for these bodies move almost in great circles, while their velocity
is great ; but about the end of their course, when that part of their appa
rent motion which arises from the parallax bears a greater proportion to
their whole apparent motion, they commonly deviate from those circles, and
when the earth goes to one side, they deviate to the other : and this deflex
ion, because of its corresponding with the motion of the earth, must arise
chiefly from the parallax ; and the quantity thereof is so considerable, as,
by my computation, to place the disappearing comets a good deal lower
than Jupiter. Whence it follows that when they approach nearer to us in
their perigees and perihelions they often descend below the orbs of Mare
and the inferior planets.
462 THE MATHEMATICAL PRINCIPLES [BOOK JJI,
The near approach of the comets is farther confirmed from the light of
r heads; for the light of a celestial body, illuminated by the sun, and
receding to remote parts, is diminished in the quadruplicate proportion of
the distance; to wit, in one duplicate proportion, on account of the increase
of the distance from the sun, and in another duplicate proportion, on ac
count of the decrease of the apparent diameter. Wherefore if both the
quantity of light and the apparent diameter of a comet are given, its dis
tance will be also given, by taking the distance of the comet to the distance
of ;i planet in the direct proportion of their diameters and the reciprocal
subduplicate proportion of their lights. Thus, in the comet of the year
1682, Mr. Flamsted observed with a telescope of 16 feet, and measured
with a micrometer, the least diameter of its head; 2 00; but the nucleus
or star in the middle of the head scarcely amounted to the tenth part of
this measure; and therefore its diameter was only 11" or 12" but in the
light and splendor of its head it surpassed that of the comet in the year
1680; and might be compared with the stars of the lirst or second magni
tude. Let us suppose that Saturn with its ring was about four times more
lucid
; and because the light of the ring was almost 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 whose diameter is 30", it follows that the distance of th
comet was to the distance of Saturn as 1 to v/4 inversely, and 12" to 30
directly ; that is, as 24 to 30, or 4 to 5. Again ; the comet in the month
of April 1665, as Hevelius informs us, excelled almost all the fixed stars
in splendor, and even Saturn itself, 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 had been compared to the stars of the
hrst magnitude. The diameter of its head was about 6
; but the nucleus,
compared with the planets by means of a telescope, was plainly less than
Jupiter ; and sometimes judged less, sometimes judged equal, to the globe
of Saturn within the ring. Since, then, the diameters of the heads of the
comets seldom exceed 8 or 12;
, and the diameter of the nucleus or central
star is but about a tenth or perhaps fifteenth part of the diameter of the
head, it appears that these stars are generally of about the same apparent
magnitude with the planets. But in regard that their light may be often
compared with the light of Saturn, yea, and sometimes exceeds it, it is evi
dent that all comets in their perihelions must either be placed below or not
far above Saturn ; and they are much mistaken who remove them almost
as far as the fixed stars ;
for if it was so, the comets could receive no more
light from our sun than our planets do from the fixed stars.
So far we have gone, without considering the obscuration which comets
suffer from that plenty of thick smoke which encompasseth their heads,
and through which the heads always shew dull, as through i cloud; for by
BoOK Hl.J Or NATURAL PHILOSOPHY. 463
how much the more a body is obscured by this smoke, by so much the more
near it must be allowed to come to the sun, that it may vie with the planeta
in the quantity of light which it reflects. Whence it is probable that
the comets descend far below the orb of Saturn, as we proved before frou
their parallax. But, above all, the thing is evinced from their tails, which
must be owing either to the sun s light reflected by a smoke arising from
them, and dispersing itself through the aether, or to the light of their own
