Figure 21

As
highlighted in "Ptolemy's Models", Ptolemy made use of Hipparchus’ model for the
Sun’s motion around the earth and calculated values for the solar
eccentricity, *e*,
and the angle *ψ*
between the line of apsides and the line joining the solstices. Figure 21 shows
the model that Ptolemy used to work out the aforesaid values and this section
aims to provide a detailed set of working to obtain the result quoted at the end
of the page: *e*
= 0.0334, *ψ*
=
12˚
58', by applying modern seasonal lengths.

Since
fall is longer than spring, EA must point somewhere between the summer solstice
and autumnal equinox. By doing so, when viewed from the earth, the sun’s orbit
between points I and J would be longer than
1/4
arc of the entire path. Then
consequently, EP
points between the winter solstice and the spring equinox. Taking the Julian
value for the number of days in a year (denoted by *y*), the mean angular
motion of the sun, *w*, is equal to
360˚/*y* per day, or

*w*
= 360˚ / 365.25 ≈ 0.9856˚ per
day

Given
that the differences in time between the sun’s arrival at G, H, I and J on its
orbit, when, viewed from the earth E, as it appears at the first points of
Capricorn, Aries, Cancer and Libra, respectively are 89^{d}0^{h},
92^{d}18^{h}, 93^{d}15^{h}, 89^{d}20^{h},
and, we work out the following angles:

These angles correspond to the seasons Winter, Spring, Summer and Autumn respectively.

As
a remark, in Heilbron’s book, the order of listing of intervals between
Capricorn, Aries, Cancer and Libra did not correspond directly with that of the
angles that followed. Had readers overlooked the mis-correspondence, and
proceeded with the calculations for *e*
and *ψ*,
errors would have been unavoidable.

Now
we continue with the working steps to obtain values for *e*
and *ψ*.
Consider triangle JSH.
Since it is an isosceles triangle, the perpendicular, of length *y*,
when dropped from point S
to the base JH,
would bisect the angle JSH.
In addition, take
*JS*
= *HS* =
*a* where
*a* is
the radius of circle centred at S.
Then,

Similarly,
in isosceles triangle GSI,
taking *x*
as the length of the perpendicular dropped from point S
to the base GI,
we have,

Due to the fact that

that is,

Hence,

By
substituting the exact values of the angles into the above equation, we obtain
the value of *y*
to be approximately 12˚59', almost equivalent to the value quoted in Heilbron’s book.

To
work out the value of eccentricity, *e*,
we note that
But

Therefore,

Using the result from above, we have,

After
substituting the relevant values into the above relation, we obtain the value of
*e* to
be around 0.0335, which is also close to that quoted in the book.

However, Ptolemy’s value of solar eccentricity which is 0.0334 exceeded that found by Kepler which is 0.0167. To account for this factor-of-two difference, we have to look into the way in which Ptolemy and Kepler measured the separation between the centre of the sun’s orbit and the earth. Figure 22 provides a visual aid to this explanation.

Figure 22

C
represents the centre of the sun’s orbit, E_{P} denotes where Ptolemy
positions the earth, E_{K} denotes where Kepler positions the earth and
X_{K} marks Kepler’s equant point. Consider radius of the sun’s
orbit to be *a*,
then the intervals between X_{K}, C, E_{K} and E_{P}
would each be
. Ptolemy calculated eccentricity as

whereas Kepler calculated the same quantity as

Thus, the factor-of-two difference is produced.

Even though we know now that Kepler’s model is the accurate one, it would be interesting to find out how Ptolemy’s model had provided such a close approximate to Kepler’s model. Since both models gave near similar observations, mathematicians and astronomers alike had to struggle with deciding which of the models was the correct one until the development of the meridiana. In fact, the use of the meridiana to make measurements that would allow substantial conclusions to be drawn from them is another area worth exploring into. In the following sections of the paper, the discussion would focus on the comparison of models and the use of the meridiana.