drawn to the earth, describes in a circular orbit.
We have above
shown that the area
which the moon de
scribes by a radius
drawn to the earth
is proportional to
the time of descrip
tion, excepting in so
far as the moon s
motion is disturbed
by the action of the
sun ; and here we
propose to investi
gate the HI equality of the moment, or horary increment of that area or
motion so disturbed. To render the calculus more easy, we shall suppose
the orbit of the moon to be circular, and neglect all inequalities but that
only which is now under consideration ; and, because of the immense dis
tance of the sun, we shall farther suppose that the lines SP and ST are
parallel. By this moans, the force LM will be always reduced to its mean
quantity TP, as well as the force TM to its mean quantity 3PK. These
forces (by Cor. 2 of the Laws of Motion) compose the force TL ; and
this force, by letting fall the perpendicular LE upon the radius TP, is
resolved into the forces TE, EL ;
of which the force TE, acting constantly
in the direction of the radius TP, neither accelerates nor retards the de
scription of the area TPC made by that radius TP ; but EL, acting on
the radius TP in a perpendicular direction, accelerates or retards the de
scription of the area in proportion as it accelerates ->r retards the moon.
BOOK lit.] OF NATURAL PHILOSOPHY.
That acceleration of the moon, in its passage from the quadrature C to the
conjunction A, is in every moment of time as the generating accclerative
3PK X TK
force EL, that is, as
.5
. Let the time be represented by the
mean motion of the moon, or (which comes to the same thing) by the angle
CTP, or even by the arc CP. At right angles upon CT erect CG equal
to CT ; and, supposing the quadrantal arc AC to be divided into an infinite
number of equal parts P/?, &c., these parts may represent the like infinite
number of the equal parts of time. Let fall pic perpendicular on CT, and
draw TG meeting with KP, kp produced in F arid /; then will FK be
equal to TK, and K/v be to PK as P/>
to T/?, that is, in a given propor-
3PK X TK
tion ; and therefore FK X K&, or the area FKkf, will be as -
~^pp >
that is, as EL: and compounding, the whole area GCKF will be as the
sum of all the forces EL impressed upon the moon in the whole time CP ;
and therefore also as the velocity generated by that sum, that is, as the ac
celeration of the description of the area CTP, or as the increment of the
moment thereof. The force by which the moon may in its periodic time
CADB of 27 1
. 7h
. 43 be retained revolving about the earth in rest at the
distance TP, would cause a body falling in the time CT to describe the
length ^CT, and at the same time to acquire a velocity equal to that with
which the moon is moved in its orbit. This appears from Cor. 9, Prop,
IV., Book I. But since K.d, drawn perpendicular on TP, is but a third
part of EL, and equal to the half of TP, or ML, in the octants, the force
EL in the octants, where it is greatest, will exceed the force ML in the
proportion of 3 to 2 ; and therefore will be to that force by which the moon
in its periodic time may be retained revolving about the earth at rest as
100 to | X 178721, or 11915; and in the time CT will generate a ve
locity equal to yylfs 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 greatest force EL in the octants be represented by the area
FK X Kk, or by the rectangle |TP X Pp, which is equal thereto; and
the velocity which that greatest force can generate in any time CP will be
to the velocity which any other lesser force EL can generate in the same
time as the rectangle |TP X CP to the area KCGF ; but the velocities
generated in the whole time CPA will be one to the other as the rectangle
2-TP X CA to the triangle TCG, or as the quadrantal arc CA to the
radius TP ; and therefore the latter velocity generated in the whole time
will be T T$TJ parts of the velocity of the moon. To this velocity of the
moon, which is proportional to the mean moment of the area (supposing
this mean moment to be represented by the number 11915), we add and
subtract the half of the other velocity ; the sum 11915 + 50, or 11965,
will represent the greatest moment of the area in the syzygy A : and the
THE MATHEMATICAL PRINCIPLES [BOOK III
difference 11915 50, or 11865, the least moment thereof in the quadra
tures. Therefore the areas which in equal times are described in the syzygies
and quadratures are one to^the other as 11965 to 11865. And if to
the least moment 11865 we add a moment which shall be to 100, the dif
ference of the two former moments, as the trapezium FKCG to the triangle
TCG, or, which comes to the same thing, as the square of the sine PK to
the square of the radius TP (that is, as Pd to TP), the sum will represent
the moment of the area when the moon is in any intermediate place P.
But these things take place only in the hypothesis that the sun and the
earth are at rest, and that the synodical revolution of the moon is finished
in 27 1
. 7h
. 43 . But since the moon s synodical period is really 29a
. 12h
.
4 T, the increments of the moments must be enlarged in the same propor
tion as the time is, that is, in the proportion of 1080853 to 1000000.
Upon whicli account, the whole increment, which was TTITTT parts of the
mean moment, will now become TY| 3- parts thereof; and therefore the
moment of the area in the quadrature of the moon will be to the moment
thereof in the syzygy as 11023 50 to 11023 + 50; or as 10973 to
11073; and to the moment thereof, when the moon is in any intermediate
place P, as 10973 to 10973 -f Pd ; that is, supposing TP = 100.
The area, therefore, which the moon, by a radius drawn to the earth,
describes m the several little equal parts of time, is nearly as the sum of
the number 219,46, and the versed sine of the double distance of the moon
from the nearest quadrature, considered in a circle which hath unity for its
radius. Thus it is when the variation in the octants is in its mean quantity.
3ut if the variation there is greater or less, that versed sine must be augnented
or diminished in the same proportion.
PROPOSITION XXVIL PROBLEM VI11.
From the horary motion of the moon to find its distance from the earth.
The area which the moon, by a radius drawn to the earth, describes in
every, moment of time, is as the horary motion of the moon and the square
of the distance of the moon from the earth conjunctly. And therefore the
distance of the moon from the earth is in a proportion compounded of the
subduplicate proportion of the area directly, and the subduplioate propor
tion of the horary motion inversely. Q.E.T.
COR. 1 . Hence the apparent diameter of the moon is given ; for it is re
ciprocally as the distance of the moon from the earth. Let astronomers
try how accurately this rule agrees with the phenomena.
COR. 2. Hence also the orbit of the moon may be more exactly defined
from the phaenomena than hitherto could be done.
BOOK III,"! OF NATURAL PHILOSOPHY. 423
PROPOSITION XXVIII. PROBLEM IX.
To find the diameters of the orbit, in which, without ec*.t itricity, the
moon would move.
The curvature of the orbit which a body describes, if attracted in lines
perpendicular to the orbit, is as the force of attraction directly, and the
square of the velocity inversely. I estimate the curvatures of lines com
pared one with another according to the evanescent proportion of the sines
or tangents of their angles of contact to equal radii, supposing those radii
to be infinitely diminished. Blit the attraction of the moon towards the
earth in the syzygies is the excess of its gravity towards the earth above
the force of the sun 2PK (see Pig. Prop. XXV); by which force the accelerative
gravity of the moon towards the sun exceeds the accelerative gravity
of the earth towards the sun, or is exceeded by it. But in the quadratures
that attraction is the sum of the gravity of the moon towards the earth,
and the sun s force KT, by which the moon is attracted towards the earth.
AT + CT 178725
And these attractions, putting N for-^-> are nearly as T^--
and - + - or as 178725N X CT* - 2000AT*
X CT, and 17S725N X AT2 + 1000CT 2 X AT. For if the accelera
tive gravity of the moon towards the earth be represented by the number
178725, the mean force ML, which in the quadratures is PT or TK, and
draws the moon towards the earth, will be 1000, and the mean force TM in
the syzygies will be 3000 ; from which, if we subtract the mean force ML,
there will remain 2000, the force by which the moon in the syzygies is
drawn from the earth, and which we above called 2 PIC. But the velocity
of the moon in the syzygies A and B is to its velocity in the quadratures
C and D as CT to AT, and the moment of the area, which the moon by
a radius drawn to the earth describes in the syzygies, to the moment of that
area described in the quadratures conjunctly ; that is, as 11073CT to
10973AT. Take this ratio twice inversely, and the former ratio once di
rectly, and the curvature of the orb of the moon in the syzygies will be to
the curvature thereof in the quadratures as 120406729 X 17S725AT 2 X
CT2 X N 120406729 X 2000AT 4 X CT to 122611329 X 178725AT2
X CT2 X N + 122611329 X 1000CT 4 X AT, that is, as 2151969AT
X CT X N 24081AT 3 to 2191371AT X CT X N + 12261CT 3
.
Because the figure of the moon s orbit is unknown, let us, in its stead,
assume the ellipsis DBCA, in the centre of which we suppose the earth to
be situated, and the greater axis DC to lie between the quadratures as the
lesser AB between the syzygies. But since the plane of this ellipsis is rerolved
about the earth by an angular motion, and the orbit, whose curva
ture we now examine, should be described in a plane void of such motion
424 THE MATHEMATICAL PRINCIPLES [BOOK III
we are to consider the figure which the moon,
while it is revolved in that ellipsis, describes iu
this plane, that is to say, the figure Cpa, the
several points p of which are found by assuming
any point P in the ellipsis, which may represent
the place of the moon, and drawing Tp equal
to TP in such manner that the angle PT/? may
be equal to the apparent motion of the sun from
the time of the last quadrature in C ; or (which
comes to the same thing) that the angle CTp
may be to the angle CTP as the time of the
synodic revolution of the moon to the time ot
the periodic revolution thereof, or as 29 1
. 12h
. 44 to 27d
. 7 1
. 43 . If, there
fore, in this proportion we take the angle CTa to the right angle CTA,
and make Ta of equal length with TA, we shall have a the lower and C
the upper apsis of this orbit Cpa. 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 described about the centre T with the interval TA, is
to the difference between the curvature of the ellipsis at the vertex A, and
the curvature of the same circle, in the duplicate proportion of the angle
CTP to the angle CTp ; and that the curvature of the ellipsis 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 described about the
centre T with the interval TC as TC to TA ; but that the curvature of
this last arch is to the curvature of the ellipsis in C in the duplicate pro
portion of TA to TC ; and that the difference betwixt the curvature of the
ellipsis in the vertex C* and the curvature of this List circle, is to the dif
ference betwixt the curvature of the figure Cpa, at the vertex C, and the
curvature of this same last circle, in the duplicate proportion of the angle
CTp to the angle CTP ;
all which proportions are easily drawn from the
sines of the angles of contact, and of the differences of those angles. But,
by comparing those proportions together, we find the curvature of the figure
Cpa at a to be to its curvature at C as AT 3,- rWoVoCT2AT to CT 3 -r
_i_6_8_2_4_AT 2 X CT ; where the number yVYVYo represents the difference
of the squares of the angles CTP and CTp, applied to the square of the
lesser angle CTP ; or (which is all one) the difference of the squares of the
limes 27. 7h
- 43 , and 29 1
. 12h
. 44 , applied to the square of the time27(1
.
7h
. 43 .
Since, therefore, a represents the syzygy of the moon, and C its quadra
ture, the proportion now found must be the same with that proportion of
the curvature of the moon s orb in the syzygies to the curvature thereof in
the quadratures, which we found above. Therefore, in order to find th
BOOK III.] OF NATURAL PHILOSOPHY. 425
proportion of CT to AT, let us multiply the extremes and the means, an(?
the terms which come out, applied to AT X CT, become 2062,79CT 4
2151969N x CT 3 + 368676N X AT X CT 2 + 36342 AT 2 X CT2 -
362047N X AT2 X CT + 2191371N X AT 3 + 4051,4AT 4 = 0.
Now if for the half sum N of the terms AT and CT we put 1, and x for
their half difference, then CT will be = 1 + x, and AT = 1 x. And
substituting those values in the equation, after resolving thereof, wre shall
find x = 0,00719 ; and from thence the semi-diameter CT = 1,00719, and
the semi-diameter AT = 0,99281, which numbers are nearly as 70^, and
692V- Therefore the moon s distance from the earth in the syzygies is to
its distance in the quadratures (setting aside the consideration of eccentrici
ty) as 09 2^ 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 figure of the moon s orbit,
partly to the inequality of the moments of the area which the moon by a
radius drawn t\) the earth describes. If the moon P revolved in the ellipsis
DBCA about the earth quiescent in the centre of the ellipsis, and by the
radius TP, drawn to the earth, described the area CTP, proportional to
the time of description ; and the greatest semi-diameter CT of the ellipsis
was to the least TA as 70 to 69; the tangent of the angle CTP would be
to the tangent of the angle of the mean motion, computed from the quad
rature C, as the semi-diameter TA of the ellipsis to its semi-diameter TC,
or as 69 to 70. But the description of the area CTP, as the moon advan
ces from the quadrature to the syzygy, ought to be in such manner accel
erated, that the moment of the area in the moon s syzygy may be to the
moment thereof in its quadrature as 11073 to 10973; and that the excess
of the moment in any intermediate place P above the moment in the quad
rature may be as the square of the sine of the angle CTP ; which we may
effect with accuracy enough, if we diminish the tangent of the angle CTP
in the subduplicate 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 octants,
where the mean motion is 45, will be found 44 27 28", which sub
tracted from 45, the angle of the mean motion, leaves the greatest varia
tion 32 32". Thus it would be, if the moon, in passing from the quad
rature to the syzygy, described an angle CTA of 90 degrees only. But
because of the motion of the earth, by which the sun is apparently trans
ferred in consequentia^ the moon, before it overtakes the sun, describes an
angle CTtf, greater than a right angle, in the proportion of the time of the
synodic revolution of the moon to the time of its periodic revolution, thai
126 THE MATHEMATICAL PRINCIPLES [BOOK III
is, in the proportion of 29 1
. 12h
. 44 to 27(l
. 7X 43 . Whence it comes tc
pass that all the angles about the centre T are dilated in the same pro
portion ; and the greatest variation, which otherwise would be but 32
32", now augmented in the said proportion, becomes 35 10".
And this is its magnitude in the mean distance of the sun from the
earth, neglecting the differences which may arise from the curvature of
the orbis magnns, and the stronger action of the sun upon the moon when
horned and new, than when gibbous and full. In other distances of the
sun from the earth, the greatest variation is in a proportion compounded
of the duplicate proportion of the time of the synodic revolution of the
moon (the time of the year being given) directly, and the triplicate pro
portion of the distance of the sun from the earth inversely. And, there
fore, in the apogee of the sun, the greatest variation is 33 14", and in its
perigee 37 11", if the eccentricity of the sun is to the transverse semi-di
ameter of the orbis magnus as 16} to 1000.
Hitherto we have investigated the variation in an orb not eccentric, in
which, to wit, the moon in its octants is always in its mean distance from
the earth. If the moon, on account of its eccentricity, is more or less re
moved from the earth than if placed in this orb, the variation may be
something greater, or something less, than according to this rule. But I
leave the excess or defect to the determination of astronomers from the
phenomena.
PROPOSITION XXX. PROBLEM XI.
To find the horary motion of the nodes of ihe moon in a circular orbit.
Let S represent the sun, T the earth, P the moon, NP/A the orbit, of thr
moon, Njo/? the orthographic projection of the orbit upon the plane of th
ecliptic : N. n the nodes. nTNm the line of the nodes produced indeti
BOOK III.] OF NATURAL PHILOSOPHY. 427
nitely ; PI, PK perpendiculars upon the lines ST, Qq ; Pp a perpendicu
lar upon the plane of the ecliptic; A, B the moon s syzygies in the plane
of the ecliptic; AZ a perpendicular let fall upon Nil, the line of the
nodes ; Q, g the quadratures of the moon in the plane of the ecliptic, and
pK a perpendicular on the line Qq lying between the quadratures. The
force of the sun to disturb the motion of the moon (by Prop. XXV) is
twofold, one proportional to the line LM, the otlier to the line MT, in the
scheme of that Proposition ; and the moon by the former force is drawn
towards the earth, by the latter towards the sun, in a direction parallel to
the right line ST joining the earth and the sun. The former force LM
acts in the direction of the plane of the moon s orbit, and therefore makes
no change upon the situation thereof, and is upon that account to be neg
lected ; the latter force MT, by which the plane of the moon s orbit is dis
turbed, is the same with the force 3PK or SIT. And this force (by Prop.
XXV) is to the force by which the moon may, in its periodic time, be uni
formly revolved in a circle about the earth at rest, as SIT to the radius of
the circle multiplied by the number 178,725, or as IT to the radius there
of multiplied by 59,575. But in this calculus, and all that follows. I
consider all the lines drawn frorri the moon to the sun as parallel to the
line which joins the earth and the sun ; because what inclination there is
almost as much diminishes all effects in some cases as it augments them
in others : and we are now inquiring after the mean motions of the nodes,
neglecting such niceties as are of no moment, and would only serve to ren
der the calculus more perplexed.
Now suppose PM to represent an arc which the moon describes in the
least moment of time, and ML a little line, the half of which the moon,
by the impulse of the said force SIT, would describe in the same time ; and
joining PL, MP, let them be produced to m and /, where they cut the plane
of the ecliptic, and upon Tm let fall the perpendicular PH. Now, since
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
those right lines lie in one common plane LMPm/, they will be parallel,
and upon that account the triangles LMP, ImP will be similar. And
seeing MPra 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 N//, which passes
through the nodes N, n, of that orbit. And because the force by which the
half of the little line LM is generated, if the whole had been together, and
it once impressed in the point P, would hav^ generated that whole line,
and caused the moon to move in the arc whoso chord is LP ;
t at is to say,
would have transferred the moon from the plane MPwT into the plane
LP/T; therefore th* angular motion of the nodes generated by that force
will be equal to the angle mTL But n.l is to raP as ML to MP ; and
since ML3
, because of the time given, is also given, ml will be as the rectan
428 THE MATHEMATICAL PRINCIPLES [BOOK III.
gle ML X mP, that is, as the rectangle IT X mP. And if Tml is a right
ano-le, the angle mTl will be as 7T^m and therefore as T^m that is (be-
ITxPH*
cause Tm and mP, TP and PH are proportional), as FFp~ and, there
fore, because TP is given, as IT X PH. But if the angle Tml or STN
is oblique, the angle mTl will be yet less, in proportion of the sine of the
angle STN to the radius, or AZ to AT. And therefore the velocity of
the nodes is as IT X PH X AZ, or as the solid content of the sines of the
three angles TPI, PTN, and STN.
If these are right angles, as happens when the nodes are in the quadra
tures, and the moon in the syzygy, the little line ml will be removed to
an infinite distance, and the angle mTl will become equal to the angle
mPl. But in this case the angle mPl is to the angle PTM, which the
moon in the same time by its apparent motion describes 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 rectilinear path; which angle, if
the gravity of the moon should have then ceased, the said force of the sun
SIT would by itself have generated 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 sun 31T should have then ceased, the force
alone by which the moon is retained in its orbit would have generated in
the same time. And. these forces (as we have above shewn) are the one to
the other as I to 59,575. Since, therefore, the mean horary motion of the
moon (in respect of the fixed stars) is 32 56" 27 "
12^-
iv
. the horary motion
of the node in this case will be 33" 10" 331V
. 12V
. But in other cases the
horary motion will be to 33" 10 " 33iv
. \2\ as the solid content of the sines
of the three angles TPI, PTN, and STN (or of the distances of the moon
from the quadrature, of the moon from the node, and of the node from the
sun) to the cube of the radius. And as often as the sine of any angle is
changed from positive to negative, and from negative to positive, so often
must the regressive be changed into a progressive, and the progressive into
a regressive motion. Whence it comes to pass that the nodes are pro
gressive as often as the moon happens to be placed between either quadra
ture, and the node nearest to that quadrature. In other cases they are
regressive, and by the excess of the regress above the progress, they are
monthly transferred in antecedentia.
COR. 1. Hence if from P and M, the extreme points of a least arc PM,
on the line Qq joining the quadratures we let fall the perpendiculars PK
MA", and produce the same till they cut the line of the nodes Nw in D ana
d, the horary motion of the nodes will be as the area MPDd, and the
square of the line AZ conjunctly. For let PK, PH, and AZ, be the three
said sines, viz., PK the sine of the distance of the moon from the quadra
BOOK III.] OF NATURAL PHILOSOPHY.
Q
421*
