Let a graduated instrument as _DAE_, (the larger the better) having a
moveable Index and Sight-holes, be fixed in such a manner, that it’s
plane surface may be parallel to the Plan of the Equator, and it’s edge
_AD_ in the Meridian: so that when the Moon is in the Equinoctial, and
on the Meridian at _E_, she may be seen through the sight-holes when the
edge of the moveable index cuts the beginning of the divisions at o, on
the graduated limb _DE_; and when she is so seen, let the _precise_ time
be noted. Now, as the Moon revolves about the Earth from the Meridian to
the Meridian again in 24 hours 48 minutes, she will go a fourth part
round it in a fourth part of that time, _viz._ in 6 hours 12 minutes, as
seen from _C_, that is, from the Earth’s center or Pole. But as seen
from _A_, the observer’s place on the Earth’s surface, the Moon will
seem to have gone a quarter round the Earth when she comes to the
sensible Horizon at _O_; for the Index through the sights of which she
is then viewed will be at _d_, 90 degrees from _D_, where it was when
she was seen at _E_. Now, let the exact moment when the Moon is seen at
_O_ (which will be when she is in or near the sensible Horizon) be
carefully noted[49], that it may be known in what time she has gone from
_E_ to _O_; which time subtracted from 6 hours 12 minutes (the time of
her going from _E_ to _L_) leaves the time of her going from _O_ to _L_,
and affords an easy method for finding the Angle _OAL_ (called _the
Moon’s horizontal Parallax_, which is equal to the Angle _ALC_) by the
following Analogy: As the time of the Moon’s describing the arc _EO_ is
to 90 degrees, so is 6 hours 12 minutes to the degrees of the Arc _DdE_,
which measures the Angle _EAL_; from which subtract 90 degrees, and
there remains the Angle _OAL_, equal to the Angle _ALC_, under which the
Earth’s Semi-diameter _AC_ is seen from the Moon. Now, since all the
Angles of a right-lined Triangle are equal to 180 degrees, or to two
right Angles, and the sides of a Triangle are always proportional to the
Sines of the opposite Angles, say, by the _Rule of Three_, as the Sine
of the Angle _ALC_ at the Moon _L_ is to it’s opposite side _AC_ the
Earth’s Semi-diameter, which is known to be 3985 miles, so is Radius,
_viz._ the Sine of 90 degrees, or of the right Angle _ACL_ to it’s
opposite side _AL_, which is the Moon’s distance at _L_ from the
observer’s place at _A_ on the Earth’s surface; or, so is the Sine of
the Angle _CAL_ to its opposite side _CL_, which is the Moon’s distance
from the Earth’s centre, and comes out at a mean rate to be 240,000
miles. The Angle _CAL_ is equal to what _OAL_ wants of 90 degrees.
[Sidenote: The Sun’s distance cannot be yet so exactly determined as the
Moon’s;