be neglected.
COR. From this and the preceding Prop, it appears that the nodes are
quiescent in their syzygies, but regressive in their quadratures, by an
hourly motion of 16" 19 " 26iv
. : and that the equation of the motion of
the nodes in the octants is 1 30 ;
all which exactly agree with the phaBnomena
of the heavens.
SCHOLIUM.
Mr. Machin, Astron., Prof. Gresh.. and Dr. Flenry Pemberton, sepa
rately found out the motion of the nodes by a different method. Mention
has been made of this method in another place. Their several papers, both
of which I have seen, contained two Propositions, and exactly agreed with
each other in both of them. Mr. Machines paper coming first to my hands,
I shall here insert it.
OF THE MOTION OF THE MOON S NODES.
< PROPOSITION I.
1 The mean motion of the sir/i from the node is defined by a geometric
mean proportional between the mean motion of the sun and that mean
motion, with which the sun recedes with the greatest swiftness from the
node in the quadratures.
" Let T be the earth s place, Nn the line of the moon s nodes at any
aiven time, KTM a perpendicular thereto, TA a right line revolving
about the centre with the same angular velocity with which the sun and
the node recede from one another, in such sort that the angle between the
quiescent right line Nra and the revolving line TA may be always equal
to the distance of the places of the sun and node. Now if any right line
TK be divided into parts TS and SK, and thost parts be taken as the
mean horary motion of the sun to the mean horary motion of the node in
the quadratures, and there be taken the right line TH, a mean propor
tional between the part TS and the whole TK, this right line will be pro
portional to the sun s mean motion from the node.
" For let there be described the circle NKnM from the centre T and
with the radius TK, and about the same centre, with the semi-axis TH
438 THE MATHEMATICAL PRINCIPLES [BOOK III
N
and TNr let there be described an ellipsis NHwL ; and in the time in
which the sun recedes from the node through the arc N0, if there be drawn
the right line Tba, the area of the sector NTa will be the exponent of the
sum of the motions of the sun and node in the same time. Let, there
fore, the extremely small arc aA. be that which the right line T/w, revolv
ing according to the aforesaid law, will uniformly describe in a given
particle of time, and the extremely small sector TAa will be as the sum
of the velocities with which the sun and node are carried two different
ways in that time. Now the sun s velocity is almost uniform, its ine
quality being so small as scarcely to produce the least inequality in the
mean motion of the nodes. The other part of this sum, namely, the mean
quantity of the velocity of the node, is increased in the recess from the
gyzygies in a duplicate ratio of the sine of its distance from the sun (by
Cor. Prop. XXXI, of this Book), and, being greatest in its quadratures
with the sun in K, is in the same ratio to the sun s velocity as SK to TS.
that is, as (the difference of the squares of TK and TH, or) the rectangle
KHM to TH 2
. But the ellipsis NBH divides the sector AT, the expo
nent of the sum of these two velocities, into two parts ABba and BTb,
proportional to the velocities. For produce BT to the circle in 0, and
from the point B let fall upon the greater axis the perpendicular BG,
which being produced both ways may meet the circle in the points F and
f; and because the space ABba is to the sector TBb as the rectangle AB
to BT2
(that rectangle being equal to the difference of the squares of TA
nnd TB, because the right line A3 is equally cut in T, and unequally in
B), therefore when the space ABba is the greatest of all in K, this ratio
will be the same as the ratio of the rectangle KHM to HT2
. But the
greatest mean velocity of the node was shewn above to be in that very
BOOK III.] OF NATURAL PHILOSOPHY. 439
ratio to the velocity of the sun ; and therefore in the quadratures the sec
tor ATa is divided into parts proportional to the velocities. And because
the rectangle KHM is to HT2 as FB/ to BG 2
, and the rectangle AB(3 is
equal to the rectangle FB/, therefore the little area ABba, where it is
greatest, is to the remaining sector TB6 as the rectangle AB/3 to BG2
But the ratio of these little areas always was as the rectangle AB# to
BT 2
; and therefore the little area ABba in the place A is less than its
correspondent little area in the quadratures in the duplicate ratio cf BG
to BT, that is, in the duplicate ratio of the sine of the sun s distance
from the node. And therefore the sum of all the little areas ABba, to
wit, the space ABN, will be as the motion of the node in the time in
which the sun hath been going over the arc NA since he left the node;
and the remaining space, namely, the elliptic sector NTB, will be as die
sun s mean motion in the same time. And because the mean annual mo
tion of the node is that motion which it performs in the time that the sun
completes one period of its course, the mean motion of the node from the
sun will be to the mean motion of the sun itself as the area of the circle
to the area of the ellipsis; that is, as the right line TK to the right line
TH, which is a mean proportional between TK and TS ; or, which comes
to the same as the mean proportional TH to the right line TS.
< PROPOSITION II.
u The rmean motion of t/ie -moon s nodes being given, to find their true
motion.
" Let the angle A be the distance of the sun from the mean place of the
node, or the sun s mean motion from the node. Then if we take the angle
B, whose tangent is to the tangent of the angle A as TH to TK, that ia,
440 THE MATHEMATICAL PRINCIPLES [BOOK JI1.
in the sub-duplicate ratio of the mean horary motion of the sun to the
mean horary motion of the sun from the node, when the node is in the
quadrature, that angle B will be the distance of the sun from the node s
true place. For join FT, and, by the demonstration of the last Propor
tion, the angle FTN will be the distance of the sun from the mean place
of the node, and the angle ATN the distance from the true place, and the
tangents of these angles are between themselves as TK to TH.
" COR. Hence the angle FTA is the equation of the moon s nodes ; and
the sine of this angle, where it is greatest in the octants, is to the radius
as KH to TK + TH. But the sine of this equation in any other place
A is to the greatest sine as the sine of the sums of the angles FTN +
ATN to the radius ;
that is, nearly as the sine of double the distance of
the sun from the mean place of the node (namely, 2FTN) to the radius.
"SCHOLIUM.
" If the mean horary motion of the nodes in the quadratures be 16"
16" 37iv
. 42V
. that is, in a whole sidereal year, 39 38 7" 50"
, TH will
be to TK in the subduplicate ratio of the number 9,0827646 to the num
ber 10,0827646, that is, as 18,6524761 to 19,6524761. And, therefore.
TH is to HK as 18,6524761 to 1
; that is, as the motion of the sun in a
sidereal year to the mean motion of the node 19 18 1" 231 ".
" But if the mean motion of the moon s nodes in 20 Julian years is
386 50 15", as is collected from the observations made use of in the
theory of the moon, the mean motion of the nodes in one sidereal year will
be 19 20 31" 58 ". and TH will be to HK as 360 to 19 20 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 16" 18 " 48iv
. And the
greatest equation of the nodes in the octants will be 1 29 57"."
PROPOSITION XXXIV. PROBLEM XV.
Tofind the horary variation of the inclination of the moon s orbit to the
plane of the ecliptic.
Let A and a represent the syzygies ; Q and q the quadratures ; N and
n the nodes ; P the place of the moon in its orbit
; p the orthographic
projection of that place upon the plane of the ecliptic ; and mTl the momentaneous
motion of the nodes as above. If upon Tm we let fall *;hc
perpendicular PG, and joining pG we produce it till it meet T/ in g, and
join also Pg~, the angle PGp will be the inclination 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 of the same after a small moment of time is elapsed;
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 conjunctly. And, therefore, if for the moment of time we as
71
Bnme an hour, since the angle GTg* (by Prop. XXX) is to the angle 33
10 " 33iv
. as IT X PG X AZ to AT 3
, the angle GP^ (or the horary va
riation of the inclination) will be to the angle 33" 10 " 33iv
. as IT X AZ
X TG X to AT 3
. Q.E.I.
And thus it would be if the moon was uniformly revolved in a circular
orbit. But if the orbit is elliptical, the mean motion of the nodes will
be diminished in proportion of the lesser axis to the greater, as we have
shewn above ; and the variation of the inclination will be also diminished
in the same proportion.
COR. 1. Upon N/i erect the perpendicular TF, and let pM. be the horary
motion of the moon in the plane of the ecliptic; upon Q.T let fall the
perpendiculars pK, MA*, and produce them till they meet TF in H and h ;
then IT will be to AT as Kk to Mjt? ; and TG to Up as TZ to AT ; and,
KA* X H# x T7
therefore, IT X TG will be equal to -=
, that is, equal to
T7
the area HpWi multiplied into the ratio ^ : and therefore the horary
variation of the inclination will be to 33" 10" 33iv
. as the area HpMA
TZ P
multiplied into AZ X ,T~ X ^ to AT 3
.
MJD PG
COR. 2. And, therefore, if the earth and nodes were after every hour
drawn back from their new and instantly restored to their old places, so as
their situation might continue given for a whole periodic month together,
the whole variation of the inclination durinor that month would be to 33
442 THE MATHEMATICAL PRINCIPLES [BOOK III
10 " 33iv
. as the aggregate of all the areas H/?MA. generated in the time ot
one revolution of the point p (with due regard in summing to their proper
P
signs + -*), multiplied into AZ X TZ X 5^ to Mjo X AT 3
; that is, as
Pp
the whole circle QAqa multiplied into AZ X TZ X *, to Mp X AT3
,
that is, as the circumference QAqa multiplied into AZ X TZ X -^ to
2Mj0 X AT2
.
COR. 3. And, therefore, in a given position of the nodes, the mean ho
rary variation, from which, if uniformly continued through the whole
month, that menstrual variation might be generated, is to 33" 10 " 33iv
. as
PD AZ x TZ
AZ X TZ X ~~ to 2AT2
, or as Pp X - LT^7p
"
to PG X 4AT; that
1 VJT - A. \
is (because Pp is to PG as the sine of the aforesaid incHnation to the ra-
AZ X TZ
dius, and - -- to 4AT as the sine of double the angle ATu to four
times the radius), as the sine of the same inclination multiplied into the
sine of double the distance of the nodes from the sun to four times the
square of the radius.
COR. 4. Seeing the horary variation of the inclination, when the nodes
are in the quadratures, is (by this Prop.) to the angle 33" 10 " 33iv
. as IT
X AZ X TG X
p
to AT 3
, that is, as *, X j~ to 2AT, that
is, as the sine of double the distance of the moon from the quadratures
Pp
multiplied into .y^
to twice the radius, the sum of all the horary varia
tions during the time that the moon, in this situation of the nodes, passes
from the quadrature to the syzygy (that is, in the space of 177} hours) will
be to the sum of as many angles 33" 10 " 331V
. or 5878 , as the sum of all
the sines of double the distance of the moon from the quadratures multi-
Pp
plied into p^ to the sum of as many diameters ; that is. as the diameter
Pp
multiplied into =~ to the circumference; that is, if the inclination be 5
1 , as 7 X i-fU* to 22> or as 27S to 1000a And>
therefore; *he whole
variation, composed out of the sum of all the horary variations in the
aforesaid time, is 103", or 2 43".
B-OGX 11I.J OF NATURAL PHILOSOPHY. 443
PROPOSITION XXXV. PROBLEM XVI.
To a given time to find the inclination of the mooiis orbit to the plant
of the ecliptic.
Let AD be the sine of the greatest inclination, and AB the sine of the
least. Bisect BD in C ; and round the centre C, with the interval BC,
describe the circle BGD. In AC take CE in the same proportion to EB
B\ HA EC
as EB to twice BA. And if to the time given we set off the angle AEG
equal to double the distance of the nodes from the quadratures, and upon
AD let fall the perpendicular GH, AH will be the sine of the inclination
required.
For GE2
is equal to GH 2 + HE2 = BHD + HE2 = HBD 4- HE2
__ BH3 = HBD + BE 2 2BH X BE = BE2 + 2EC X BH = SEC
X AB + 2EC X BH=2EC X AH; wherefore since 2EC is given. GE2
will be as AH. Now let AEg- represent double the distance of the nodes
from the quadratures, in a given moment of time after, and the arc G^, on
account of the given angle GE^-, will be as the distance GE. But HA is
to GO- as GH to GC, and, therefore, HA is as the rectangle GH X G^, or
GH x GE, that is, as ^ X GE2
, or 7^ X AH: that is, as AH and
ljr_ti
the sine of the angle AEG conjunctly. If, therefore, in any one case. AH
be the sine of inclination, it will increase by the same increments as the
bine of inclination doth, by Cor. 3 of the preceding Prop, and therefore will
always continue equal to that sine. But when the point G falls upon
Cither point B or D, AH is equal to this sine, and therefore remains always
equal thereto. Q.E.D.
In this demonstration I have supposed that the angle BEG, representing
double the distance of the nodes from the quadratures, increaseth uniform
ly ;
for I cannot descend to every minute circumstance of inequality. Now
suppose that BEG is a right angle, and that Gg is in this case the ho
rary increment of double the distance of the nodes from the sun ; then, by
Cor. 3 of the last Prop, the horary variation of the inclination in the same
case will be to 33" 10" 33iv
. as the rectangle of AH, the sine of the incli
nation, into the sine of the right angle BEG, double the distance of the
nodes from the sun, to four times the square of the radius ;
that is, as AH,
THE MATHEMATICAL PRINCIPLES [Bc-OK )lL
the sine of the mean inclination, to four times the radius; that is, seeing
the mean inclination is about 5 S, as its sine 896 to 40000, the quad
ruple of the radius, or as 224 to 10000. But the whole variation corres
ponding to BD, the difference of the sines, is to this horary variation as
the diameter BU to the arc G%, that is, conjunctly as the diameter BD to
the semi- circumference BGD, and as the time of 2079 T\ hours, in which
the node proceeds from the quadratures to the syzyffies, to one hour, that
is as 7 to 11, and 2079 T\ to 1. Wherefore, compounding all these pro
portions, we shall have the whole variation BD to 33" 10" 33iv
. as 224 X
7 X 2079 T\ to 110000, that is, as 29645 to 1000; and from thence that
variation BD will come out 16 23i".
And this is the greatest variation of the inclination, abstracting from
the situation of the moon in its orbit: for if the nodes are in the syzygies,
the inclination suffers no change from the various positions of the moon.
But if the nodes are in the quadratures, the inclination is less when the
moon is in the syzygies than when it is in the quadratures by a difference
of 2 43", as we shewed in Cor. 4 of the preceding Prop. ; and the whole
mean variation BD, diminished by 1 21 i", the half of this excess, becomes
15 2", when the moon is in the quadratures: and increased by the same,
becomes 17 45" when the moon is in the syzygies. If, therefore, the
moon be in the syzygies, the whole variation in the passage of the nodes
from the quadratures to the syzygies will be 17 45"
; and, therefore, if the
inclination be 5 17 20", when the nodes are in the syzygies, it will be 4
59 35" when the nodes are in the quadratures and the moon in the syzy
gies. The truth of all which is confirmed by observations.
Now if the inclination of the orbit should be required when the moon is
in the syzygies, and the nodes any where between them and the quadratures,
let AB be to AD as the sine of 4 59 35" to the sine of 5 17 20", and
take the angle AEG equal to double the distance of the nodes from the
quadratures ; and AH will be the sine of the inclination desired. To this
inclination of the orbit the inclination of the same is equal, when the moon
is 90 distant from the nodes. In other situations of the moon, this men
strual 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 menstrual inequality of the motion of the nodes (as we said
before), and therefore may be neglected in the computation of the said
latitude.
SCHOLIUM.
By these computations of the lunar motions I was willing to shew that
by the theory of gravity the motions of the moon could be calculated from
their physical causes. By the same theory I moreover found that the an
nual equation of the mean motion of the moon arises from the various
BOOK III.] OF NATURAL PHILOSOPHY 445
dilatation which the orbit of the moon suffers from the action of the sun
according to Cor. 6, Prop. LXVI. Book I. The force of this action is
greater in the perigeon sun, and dilates the moon s orbit
; in the apogeon
sun it is less, and permits the orbit to be again contracted. The moon
moves slower in the dilated and faster in the contracted orbit ; and the
annual equation, by which this inequality is regulated, vanishes in the
apogee and perigee of the sun. In the mean distance of the sun from the
earth it arises to about 11 50"
; in other distances of the sun it is pro
portional to the equation of the sun s centre, and is added to the mean
motion of the moon, while the earth is passing .from its aphelion to its
perihelion, and subducted while the earth is in the opposite semi-circle.
Taking for the radius of the orbis niagnus 1000, and 16} for the earth s
eccentricity, this equation, when of the greatest magnitude, by the theory
of gravity comes out 11 49". But the eccentricity of the earth seems to
be something greater, and with the eccentricity this equation will be aug
mented in the same proportion. Suppose the eccentricity 16}^, and the
greatest equation will be 11 51".
Farther ;
I found that the apogee and nodes of the moon move fastei
in the perihelion of the earth, where the force of the sun s action is greater,
than in the aphelion thereof, and that in the reciprocal triplicate propor
tion of the earth s distance from the sun ; and hence arise annual equa
tions of those motions proportional to the equation of the sun s centre.
Now the motion of the sun is in the reciprocal duplicate proportion of the
earth s distance from the sun ; and the- greatest equation of the centre
which this inequality generates is 1 56 20", corresponding to the abovementioned
eccentricity of the sun, 16}. But if the motion of the sun
had been in the reciprocal triplicate proportion of the distance, this ine
quality would have generated the greatest equation 2 54 30"
; and there
fore the greatest equations which the inequalities of the motions of the
moon s apogee and nodes do generate are to 2 54 30" as the mean diur
nal motion of the moon s apogee and the mean diurnal motion of its
nodes are to the mean diurnal motion of the sun. Whence the greatest
equation of the mean motion of the apogee comes out 19 43", and the
greatest equation of the mean motion of the nodes 9 24". The former
equation is added, and the latter subducted, while the earth is passing
from its perihelion to its aphelion, and contrariwise when the earth is in
the opposite semi-circle.
By the theory of gravity I likewise found that the action of the sun
upon the moon is something greater when the transverse diameter of the
moon s orbit passeth through the sun than when the same is perpendicu
lar upon the line which joins the earth and the sun ; and therefore the
moon s orbit is something larger in the former than in the latter case.
And hence arises another equation of the moon s moan motion, depending
446 THE MATHEMATICAL PRINCIPLES [BOOK III
upon the situation of the moon s apogee in respect of the sun, which is in
its greatest quantity when the moon s apogee is in the octants of the sun,
and vanishes when the apogee arrives at the quadratures or syzygies ; and
it is added to the mean motion while the moon s apogee is passing from
the quadrature of the sun to the syzygy, and subducted while the apogee
is passing from the syzygy to the quadrature. This equation, which I
shall call the semi-annual, when greatest in the octants of the apogee,
arises to about 3 45", so far as I could collect from the phenomena : and
this is its quantity in the mean distance of the sun from the earth. But
it is increased and diminished in the reciprocal triplicate proportion of
the sun s distance, and therefore is nearly 3 34" when that distance is
greatest^ and 3 56" when least. But when the moon s apogee is without
the octants, it becomes less, and is to its greatest quantity as the sine of
double the distance of the moon s apogee from the nearest syzygy or quad
rature to the radius.
By the same theory of gravity, the action of the sun upon the moon is
something greater when the line of the moon s nodes passes through the
sun than when it is at right angles with the line which joins the sun and
the earth ; and hence arises another equation of the moon s mean motion,
which I shall call the second semi-annual ; and this is greatest when the
nodes are in the octants of the sun, and vanishes when they are in the
syzygies or quadratures ; and in other positions of the nodes is propor
tional to the sine of double the distance of either node from the nearest
syzygy or quadrature. And it is added to the mean motion of the moon,
if the sun is in antecedentia, to the node which is nearest to him, and
