Let us now suppose that the Radius of the circular curve described by
the nail in the axle is to the Radius of the Circle which the pin in the
nave describes round the axle as 337-1/2 to 1; which is the proportion
of the Radius or Semidiameter of the Earth’s Orbit to that of the
Moon’s; or of the circular curve _A_ 1 2 3 4 5 6 7 _B_ &c. to the little
Circle _a_; and then, whilst the progressive nail describes the said
curve from _A_ to _E_, the pin will go once round the nail with regard
to the center of it’s path, and in doing so, will describe the curve
_abcde_. The former will be a true representation of the Earth’s path
for one Lunation, and the latter of the Moon’s for that time. Here we
may set aside the inequalities of the Moon’s Moon, and also the Earth’s
moving round it’s common center of gravity and the Moon’s: all which, if
they were truly copied in this experiment, would not sensibly alter the
figure of the paths described by the nail and pin, even though they
should rub against a plain upright surface all the way, and leave their
tracks visible. And if the chariot should be driven forward on such a
convex piece of ground, so as to turn the wheel several times round, the
track of the pin in the nave would still be concave toward the center of
the circular curve described by the pin in the Axle; as the Moon’s path
is always concave to the Sun in the center of the Earth’s annual Orbit.
[Sidenote: Proportion of the Moon’s Orbit to the Earth’s.]
In this Diagram, the thickest curve line _ABCD_, with the numeral
figures set to it, represents as much of the Earth’s annual Orbit as it
describes in 32 days from west to east; the little Circles at _a_, _b_,
_c_, _d_, _e_ shew the Moon’s Orbit in due proportion to the Earth’s;
and the smallest curve _abcdef_ represents the line of the Moon’s path
in the Heavens for 32 days, accounted from any particular New Moon at
_a_. The machine, Fig. 5th is for delineating the Moon’s path, and will
be described, with the rest of my Astronomical machinery, in the last
Chapter. The Sun is supposed to be in the center of the curve _A 1 2 3 4
5 6 7 B_ &c. and the small dotted Circles upon it represent the Moon’s
Orbit, of which the Radius is in the same proportion to the Earth’s path
in this scheme, that the Radius of the Moon’s Orbit in the Heavens bears
to the Radius of the Earth’s annual path round the Sun; that is, as
240,000 to 81,000,000, or as 1 to 337-1/2.
[Sidenote: Fig. II.]
When the Earth is at _A_ the New Moon is at _a_; and in the seven days
that the Earth describes the curve _1 2 3 4 5 6 7_, the Moon in
accompanying the Earth describes the curve _ab_; and is in her first
Quarter at _b_ when the Earth is at _B_. As the Earth describes the
curve _B 8 9 10 11 12 13 14_ the Moon describes the curve _bc_; and is
opposite to the Sun at _c_, when the Earth is at _C_. Whilst the Earth
describes the curve _C 15 16 17 18 19 20 21 22_ the Moon describes the
curve _cd_; and is in her third Quarter at _d_ when the Earth is at _D_.
Once more, whilst the Earth describes the curve _D 23 24 25 26 27 28 29_
the Moon describes the curve _de_; and is again in conjunction at _e_
with the Sun when the Earth is at _E_, between the 29th and 30th day of
the Moon’s age, accounted by the numeral Figures from the New Moon at
_A_. In describing the curve _abcde_, the Moon goes round the
progressive Earth as really as if she had kept in the dotted Circle _A_,
and the Earth continued immoveable in the center of that Circle.
[Sidenote: The Moon’s motion always concave towards the Sun.]
And thus we see, that although the Moon goes round the Earth in a
Circle, with respect to the Earth’s center, her real path in the Heavens
is not very different in appearance from the Earth’s path. To shew that
the Moon’s path is concave to the Sun, even at the time of Change, it is
carried on a little farther into a second Lunation, as to _f_.
[Sidenote: How her motion is alternately retarded and accelerated.]
267. The Moon’s absolute motion from her Change to her first Quarter, or
from _a_ to _b_, is so much slower than the Earth’s, that she falls 240
thousand miles (equal to the Semidiameter of her Orbit) behind the Earth
at her first Quarter in _b_, when the Earth is in _B_; that is, she
falls back a space equal to her distance from the Earth. From that time
her motion is gradually accelerated to her Opposition or Full at _c_,
and then she is come up as far as the Earth, having regained what she
lost in her first Quarter from _a_ to _b_. From the Full to the last
Quarter at _d_ her motion continues accelerated, so as to be just as far
before the Earth at _D_, as she was behind it at her first Quarter in
_b_. But, from _d_ to _e_ her motion is retarded so, that she loses as
much with respect to the Earth as is equal to her distance from it, or
to the Semidiameter of her Orbit; and by that means she comes to _e_,
and is then in conjunction with the Sun as seen from the Earth at _E_.
Hence we find, that the Moon’s absolute motion is slower than the
Earth’s from her third Quarter to her first; and swifter than the
Earth’s from her first Quarter to her third: her path being less curved
than the Earth’s in the former case, and more in the latter. Yet it is
still bent the same way towards the Sun; for if we imagine the concavity
of the Earth’s Orbit to be measured by the length of a perpendicular
line _Cg_, let down from the Earth’s place upon the straight line _bgd_
at the Full of the Moon, and connecting the places of the Earth at the
end of the Moon’s first and third Quarters, that length will be about
640 thousand miles; and the Moon when New only approaching nearer to the
Sun by 240 thousand miles than the Earth is, the length of the
perpendicular let down from her place at that time upon the same
straight line, and which shews the concavity of that part of her path,
will be about 400 thousand miles.
[Sidenote: A difficulty removed.