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 89d0h, 92d18h, 93d15h, 89d20h, 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, EP denotes where Ptolemy
positions the earth, EK denotes where Kepler positions the earth and
XK marks Kepler’s equant point. Consider radius of the sun’s
orbit to be a,
then the intervals between XK, C, EK and EP
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.