ture, PH the sine of the distance of the moon from the node, and AZ the
of the distance of the node from the sun ; and the velocity of the node
will be as the solid content of PK X PH X AZ. But PT is to PK as
PM to KA;; and, therefore, because PT and PM are given, Kk will be as
PK. Likewise AT is to PD as AZ to PH, and therefore PH is as the
rectangle PD X AZ ; and, by compounding those proportions, PK X PH
is as the solid content Kk X PD X AZ; and PK X PH X AZ as KA
X PD X AZ 2
; that is, as the area PDrfM and AZ 3
conjunctly. Q.E.I).
COR. 2. In any given position of the nodes their mean horary motion is
half their horary motion in the moon s syzygies ; and therefore is to 16"
35 " 16iv
. 36V
. as the square of the sine of the distance of the nodes from
the syzygies to the square of the radius, or as AZ 2 to AT2
. For if the
moon, by an uniform motion, describes the semi-circle QA</, the sum of all
the areas PDdM, during the time of the moon s passage from Q, to M, will
make up the area QMc/E. terminating at the tangent Q,E of the circle ;
and by the time that the moon has arrived at the point //, that sum will
make up the whole area EQ,Aw described by the line PD : but when the
moon proceeds from n to q, the line PD will fall without the circle, and
describe the area nqe, terminating at the tangent qe of the circle, which
area, because the nodes were before regressive, but are now progressive,
must be subducted from the former area, and, being itself equal to the area
Q.EN, will leave the semi-circle NQAn. While, therefore, the moon de
scribes a semi-circle, the sum of all the areas PDdM will be the area of
that semi-circle ; and while the moon describes a complete circle, the sum
of those areas will be the area of the whole circle. But the area PDc^M,
when the moon is in the syzygies, is the rectangle of the arc PM into the
radius PT ; and the sum of all the areas, every one equal to this area, in
the time that the moon describes a complete circle, is the rectangle of the
whole circumference into the radius of the circle
; and this rectangle, being
double the area of the circle, will be double the quantity of the former sum
130 THE MATHEMATICAL PRINCIPLES [BOOK 1IJ
If, therefore, the nodes went on with that velocity uniformly continued
which they acquire in the moon s syzygies, they would describe a space
double of that which they describe in fact
; and, therefore, the mean motion,
by which, if Uniformly continued, they would describe the same space with
that which they do in fact describe by an unequal motion, is but one-half
of that motion which they are possessed of in the moon s syzygies. Where
fore since their greatest horary motion, if the nodes are in the quadratures,
is 33" 10 " 33iv
. 12V
. their mean horary motion in this case will be 16"
35 " 16iv
. 36V
. And seeing the horary motion of the nodes is every where
as AZ 2 and the area PDdM conjunctly, and. therefore, in the moon s
syzygies, the horary motion of the nodes is as AZ 2 and the area PDdM
conjunctly, that is (because the area PDdNL described in the syzygies is
given), as AZ2
, therefore the mean motion also will be as AZ2
; and, there
fore, when the nodes are without the quadratures, this motion will be to
16" 35 " I6 v
. 36V
. as AZ 2 to AT 2
. Q.E.D.
PROPOSITION XXXI. PROBLEM XII.
To find the horary motion of the nodes of the moon in an elliptic orbit
Let Qjpmaq represent an ellipsis described with the greater axis Qy, am
the lesser axis ab : QA^B a circle circumscribed ; T the earth in the com
mon centre of both ; S the sun ; p the moon moving in this ellipsis ; and
BOOK I1I.J OF NATURAL PHILOSOPHY. 431
pm an arc which it describes in the least moment of time; N and n tlw
nodes joined by the line N//, ; pK and ink perpendiculars upon the axis
Q,</,
produced both ways till they meet the circle in P and M, and the line of
the nodes in D and cl. And if the moon, by a radius drawn to the earth,
describes an area proportional to the time of description, the horary motion
of the node in the ellipsis will be as the area pDdm and AZ2
conjunctly.
For let PF touch the circle in P, and produced meet TN in F; arid pj
touch the ellipsis in p, and produced meet the same TN in /, and both
tangents concur in the axis TQ, at Y. And let ML represent the space
which the moon, by the impulse of the above-mentioned force 3IT or 3PK,
would describe with a transverse motion, in the meantime while revolving
in the circle it describes the arc PM ; and ml denote the space which the
moon revolving in the ellipsis would describe in the same time by the im
pulse of the same force SIT or 3PK ; and -let LP and Ip be produced till
they meet the plane of the ecliptic in G and g, and FG and /"^
be joined,
of which FG produced may cut pf, pa; and TQ, in c, e, and R respect
ively ; and/0" produced may cut TQ in r. Because the force SIT or 3PK
in the circle is to the force SIT or 3/?K in the ellipsis as PK to /?K, or
as AT to T, the space ML generated by the former force will be to the
space ml generated by the latter as PK to
p"K ;
that is, because of the
similar figures PYK/? and FYRc, as FR to cR. But (because of the
similar triangles PLM, PGF) ML is to FG as PL to PG. that is (on ac
count of the parallels L/r, PK, GR), as pi to pe, that is (because of the
similar triangles plm, cpe), as lm to ce ; and inversely as LM is to lm, or
as FR is to cR, so is FG to ce. And therefore if fg was to ce as/// to
cY, that is, as fr to cR (that is, as fr to FR and FR to cR conjunctly,
that is, as/T to FT, and FG to ce conjunctly), because the ratio of FG
to ce, expunged on both sides, leaves the ratiosfg to FG and/T to FT,
fg would be to FG as/T to FT; and, therefore, the angles which FG
and/- would subtend at the earth T would be equal to each other. But
these angles (by what we have shewn in the preceding Proposition) are the
motions of the nodes, while the moon describes in the circle the arc PM,
in the ellipsis the arc jt?w; and therefore the motions of the nodes in the
circle and in the ellipsis would be equal to each other. Thus, I say, it
cex /Y
would be, if fg was to cc as/Y to cY, that is, if/,r was oqual to ^ .
But because of the similar triangles/?/?, cep, fg is to cc as//? to cp ; anJ
therefore/?- is equal to -
; and therefore the angle which fg sub
tends in fact is to the former angle which FG subterds. that is to say, the
motion of the nodes in tl;^ ellipsis is to the motion of the same in the
circle aa this/^ or- to the forrer/o- or , that is, as//? X
432 THE MATHEMATICAL PRINCIPLES [HOOK 111.
cY to/ Y X cp, or as//? to/ Y, and cY to cjo ; that is;
if ph parallel to
TN meet FP in h, as FA to FY and FY to FP ; that is, as Fh to FP
or DJO to DP, and therefore as the area Dpmd to the area DPMc?. And,
therefore, seeing (by Corol. 1, Prop. XXX) the latter area and AZ2 conjunctly
are proportional to the horary motion of the nodes in the circle,
the former area and AZ2
conjunctly will be proportional to the horary
motion of the nodes in the ellipsis. Q.E.D.
COR. Since, therefore, in any given position of the nodes, the sum of all
the areas />Drfm, in the time while the moon is carried from the quadra
ture to any place tn, is the area mpQ&d terminated at the tangent of the
ellipsis Q,E ; and the sum of all those areas, in oTne entire revolution, is
the area of the whole ellipsis ; the mean motion of the nodes in the ellip
sis will be to the mean motion of the nodes in the circle as the ellipsis to
the circle
; that is, as Ta to TA, or 69 to 70. And, therefore, since (by
Corol 2, Prop. XXX) the mean horary motion of the nodes in the circle
is to 16" 35"
7 16iv
. 36V
. as AZ2 to AT2
, if we take the angle 16" 21 "
3iv
. 30V
. to the angle 16" 35 " 16iv
. 36V
. as 69 to 70. the mean horary mo
tion of the nodes in the ellipsis will be to 16" 21 " 3iv
. 30V
. as AZ2 to
AT2
;
that is, as the square of the sine of the distance of the node from
the sun to the square of the radius.
But the moon, by a radius drawn to the earth, describes the area in the
syzygie-s with a greater velocity than it does that in the quadratures, and
upon that account the time is contracted in the syzygies, and prolonged in
the quadratures ; and together with the time the motion of the nodes is
likewise augmented or diminished. But the moment of the area in the
quadrature of the moon was to the moment thereof in the syzygies as
10973 to 11073 ; and therefore the mean moment in the octants is to the
excess in the syzygies. and to the defect in the quadratures, as 1 1023, the
half sum of those numbers, to their half difference1 50. Wherefore since
the time of the moon in the several little equal parts of its orbit is recip
rocally as its velocity, the mean time in the -octants will be to the excess
of the time in the quadratures, and to the defect of the lime in the syzy
gies arising from this cause, nearly as 11023 to 50. But, reckoning from
the quadratures to the syzygies, I find that the excess of the moments of
the area, in the several places above the least moment in the quadratures,
is nearly as the square of the sine of the moon s distance from the quad
ratures : and therefore the difference betwixt the moment in any place,
and the mean moment in the octants, is as the difference betwixt the square
of the sine of the moon s distance from the quadratures, and the square
of the sine of 45 degrees, or half the square of the radius ; and the in
crement of the time in the several places between the octants and quad
ratures, and the decrement thereof between the octants and syzygies, is in
the same proportion. But the motion of the nodes, while the moon de
scribes the several little equal parts of its orbit, is accelerated or retarded
BOOK III.] OF NATURAL PHILOSOPHY. 433
in the duplicate proportion of the time ; for that motion, while the moou
describes PM, is (cceteris parilms) as ML. and ML is in the duplicate
proportion of the time. Wherefore the motion of the nodes in the syzygj-
es, in the time while the moon describes given little parts of its orbit,
is diminished in the duplicate proportion of the number H07. J to the num
ber 11023: and the decrement is to the remaining motion as 100 to
10973 ; but to the whole motion as 100 to 11073 nearly. But the decre
ment in the places between the octants and syzygies, and the increment in
the places between the octants and quadratures, is to this decrement nearly
as the whole motion in these places to the whole motion in the syzygies,
and the difference betwixt the square of the sine of the moon s distance
from the quadrature, and the half square of the radius, to the half square
of the radius conjunctly. Wherefore, if the nodes are in the quadratures,
and we take two places, one on one side, one on the other, equally distant
from the octant and other two distant by the same interval, one from the
syzygy, the other from the quadrature, and from the decrements of the
motions in the two places between the syzygy and octant we subtract the
increments of the motions in the two other places between the octant and
the quadrature, the remaining decrement will be equal to the decre
ment in the syzygy, as will easily appear by computation ; and therefore
the mean decrement, which ought to be subducted from the mean motion
of the nodes, is the fourth part of the decrement in the syzygy. The
whole horary motion of the nodes in the syzygies (when the moon by a ra
dius drawn to the earth was supposed to describe an area proportional to
the time) was 32" 42" ? iv
. And we have shewn that the decrement of
the motion of the nodes, in the time while the moon, now moving with
greater velocity, describes the same space, was to this motion as 100 to
1.1073; and therefore this decrement is 17 " 43iv
. 11 v
. The fourth part
of which 4 " 25iv
. 48V
. subtracted from the mean horary motion above
found, 16" 21 //; 3iv
. 30V
. leaves 16" 16 " 37iv
. 42V
. their correct mean ho
rary motion.
If the nodes are without the quadratures, and two places are considered,
one on one side, one on the other, equally distant from the syzygies, the
sum of the motions of the nodes, when the moon is in those places, will be
to the sum of their motions, when the moon is in the same places and the
nodes in the quadratures, as AZ2 to AT2
. And the decrements of the
motions arising from the causes but now explained will be mutually as
the motions themselves, and therefore the remaining motions will be mu
tually betwixt themselves as AZ2 to AT2
; and the mean motions will be
as the remaining motions. And, therefore, in any given position of the
nodes, their correct mean horary motion is to 16" 16 " 37iv
. 42V
. as AZ2
to AT2
; that is, as the square of the sine of the distance of the nodes
from the syzygies to the square of the radius.
28
434 THE MATHEMATICAL PRINCIPLES [BOOK III
PROPOSITION XXXII. PROBLEM XIII.
Tofind the mean motion of the nodes of the moon.
The yearly mean motion is the sum of all the mean horary motions
throughout the course of the year. Suppose that the node is in N, and
that, after every hour is elapsed, it is drawn back again to its former
place; so that, notwithstanding its proper motion, it may constantly re
main in the same situation with respect to the fixed stars; while in the
mean time the sun S, by the motion of the earth, is seen to leave the node,
and to proceed till it completes its appa
rent annual course by an uniform motion.
Let Aa represent a given least arc, which
the right line TS always drawn to the
sun, by its intersection with the circle
NA?/, describes in the least given moment
of time; and the mean horary motion
(from what we have above shewn) will be
as AZ 2
, that is (because AZ and ZY are
proportional), as the rectangle of AZ into ZY. that is, as the area
AZYa ; and the sum of all the mean horary motions from the beginning
will be as the sum of all the areas oYZA, that is, as the area NAZ. But
the greatest AZYa is equal to the rectangle of the arc Aa into the radius
of the circle ; and therefore the sum of all these rectangles in the whole
circle will be to the like sum of all the greatest rectangles as the area of
the whole circle to the rectangle of the whole circumference into the ra
dius, that is, as 1 to 2. But the horary motion corresponding to that
greatest rectangle was 16" 16 " 37iv
. 42V
. and this motion in the complete
course of the sidereal year, 365d
. 6". 9 , amounts to 39 38 7" 50"
, and
therefore the half thereof, 19 49 3" 55"
,
is the mean motion of the
nodes corresponding to the whole circle. And the motion of the nodes,
in the time while the sun is carried from N to A, is to 19 49 3" 55" as
the area NAZ to the whole circle.
Thus it would be if the node was after every hour drawn back again to
its former place, that so, after a complete revolution, the sun at the year s
end would be found again in the same node which it had left when the
year begun. But, because of the motion of the node in the mean time, the
sun must needs meet the node sooner ; and now it remains that we compute
the abbreviation of the time Since, then, the sun, in the course of the
year, travels 360 degrees, and the node in the same time by its greatest
motion would be carried 39 > 38 7" 50 ",
or 39,6355 degrees ; and the mean
motion of the node in any place N is to its mean motion in its quadratures
as AZ 2 to AT- the motion of the sun will be to the motion of the noda
BOOK III.] OF NATURAL PHILOSOPHY. 43o
in N as 360AT2 to 39,6355AZ2
; that is, as 9,OS27646AT 2 to AZ .
Wherefore if we suppose the circumference NA/* of the whole circle to he
divided into little equal parts, such as Aa, the time in which the sun would
describe the little arc Aa, if the circle was quiescent, will be to the time of
which it would describe the same arc, supposing the circle together with
the nodes to be revolved about the centre T, reciprocally as 9,0827646AT2
to 9,082764 6AT2 + AZ 2
;
for the time is reciprocally as the velocity
with which the little arc is described, and this velocity is the sum of the
velocities of both sun and node. If, therefore, the sector NTA represent
the time in which the sun by itself, without the motion of the node, would
describe the arc NA, and the indefinitely small part ATa of the sector
represent the little moment of the time in which it would describe the least
arc Aa ; and (letting fall aY perpendicular upon N//) if in AZ we take
c/Z of such length that the rectangle of dZ into ZY may be to the least
part AT of the sector as AZ 2 to 9,OS27646AT 2 -f AZ 2
, that is to
say, that dZ may be to |AZ as AT2 to 9,0827646AT 2
-f AZ 2
;
the
rectangle of dZ into ZY will represent the decrement of the time arising
from the motion of the node, while the arc Aa is described ; and if the
curve NdGn is the locus where the point d is always found, the curvilinear
area Ne/Z will be as the whole decrement of time while the whole arcNA
is described
; and; therefore, the excess of the sector NAT above the area
NrfZ will be as the whole time But because the motion of the node in a
less time is less in proportion of the time, the area AaYZ must also be di
minished in the same proportion : which may be done by taking in AZ the
line eZ of such length, that it may be to the length of AZ as AZ 2 to
9,OS27646AT 2 -f AZ 2
; for so the rectangle of eZ into ZY will be to
the area AZYa as the decrement of the time in which the arc Aa is de
scribed to the whole time in which it would have been described, if the
node had been quiescent ; and, therefore, that rectangle will be as the de
crement of the motion of the node. And if the curve NeFn is the locus of
the point e, the whole area NeZ, which is the sum of all the decrements of
that motion, will be as the whole decrement thereof during the time in
which the arc AN is described ; and the remaining area N Ae will be as the
remaining motion, which is the true motion of the node, during the time
in which the whole arc NA is described by the joint motions of both sun
and node. Now the area of the semi-circle is to the area of the figure
NeFn found by the method of infinite series nearly as 793 to o-\ But the
motion corresponding or proportional to the whole circle was 19 49 3"
55 "
; and therefore the motion corresponding- to double the figure NeF//
is t 29 58" 2 ", which taken from the former motion leaves 18 19 5"
53 ", the whole motion of the node witn respect to the fixed stars in the
interval between two of its conjunction? with the sun ; and this motion sub
ducted from the annual motion of the sun 360C
, leaves 341 40 54" 7 ",
4o6 THE MATHEMATICAL PRINCIPLES [BOOK 111.
the motion of the sun in the interval between the same conjunctions. But
as this motion is to the annual motion 360, so is the motion of the node
but just now found 1S 19 5" 53 " to its annual motion, which will there
fore be 19 IS I" 23 "
; and this is the mean motion of the nodes in the
sidereal year. By astronomical tables, it is 19 21 21" 50 ". The dif
ference is less than 3- j^- part of the whole motion, and seems to arise 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 inclination of the orbit,
the motion of the nodes is something retarded, and reduced to its just
velocity.
PROPOSITION XXXIII. PROBLEM XIV.
To find the true motion, of the nodes of the moon.
In the time which is as the area
NTA NrfZ (in the preceding Fig.)
that motion is as the area NAe, and
is thence given ; but because the cal
culus is too difficult, it will be better
to use the following construction of
the Problem. About the centre C,
with any interval CD, describe the circle BEFD ; produce DC to A so as
AB may be to AC as the mean motion to half the mean true motion when
the nodes are in their quadratures (that is, as 19 18 I" 23 " to 19 49 3"
55 "
; and therefore BC to AC as the difference of those motions G Jl 2"
32 " to the latter motion 19 49 3" 55 ", that is, as 1 to 38 T\). Then
through the point D draw the indefinite line Gg, touching the circle in.
I)
; and if we take the angle BCE, or BCF, equal to the double distance
of the sun from the place of the node, as found by the mean motion, and
drawing AE or AF cutting the perpendicular DG in G, we take another
angle which shall be to the whole motion of the node in the interval be
tween its syzygies (that is, to 9 IV 3") as the tangent DG to the whole
circumference of the circle BED, and add this last angle (for which the
angle DAG may be used) to the mean motion of the nodes, while they are
passing from the quadratures to the syzygies, and subtract it from their
mean motion while they are passing from the syzygies to the quadratures,
we shall have their true motion ; for the true motion so found will nearly
agree with the true motion which comes out from assuming the times as
the area NTA NrfZ, and the motion of the node as the area NAe ;
as
whoever will please to examine and make the computations will find : and
this is the semi -menstrual equation of the motion of the nodes. But there
is also a menstrual equation, but which is by no means necessary for find
BOOK III.] OF NATURAL PHILOSOPHY. 437
ing of the moon s latitude ;
for since the variation of the inclination of the
moon s orbit to the plane of the ecliptic is liable to a twofold inequality,
the one semi-menstrual, the other menstrual, the menstrual inequality of
this variation, and the menstrual equation of the nodes, so moderate and
carrect each other, that in computing the latitude of the moon both may
