There had been a great controversy between adherents of Ptolemy’s traditional solar theory and proponents of Kepler’s “bisection of the eccentricity” in his solar model. This motivated Cassini to find out which theory was the accurate one by using the meridiana at San Petronio; and he succeeded. The basis of and working steps to his conclusion are elaborated in this section.
As highlighted in "Ptolemy's Solar Eccentricity", the modern value of the eccentricity is half of the ancient. Suppose in Ptolemy’s model, the solar eccentricity is of value e, then the same quantity would be of value in Kepler’s model.
Figure 25
Referring to Figure 25(a), we see that at perigee, the distance between Sun and Earth is shorter on Ptolemy’s model than that on Kepler’s. From 25(b), the opposite occurs: at apogee, the same distance is longer on Ptolemy’s model than that on Kepler’s. If the apsidal separation between Sun and Earth could be observed directly, then it would be possible to test out which of the two models is the correct one. Such a direct observation cannot be obtained but fortunately, a convenient substitute exists in the Sun’s apparent diameter, which is inversely proportional to its separation from the Earth. Astronomers had found that the Sun’s apparent diameter at mean distance ranges from 30' 30" to 32' 44". In addition, the difference between the Sun’s diameters at the absides was found to be either 1' or 2'. The table below lists the apogee distance, perigee distance and apsidal difference between Sun and Earth corresponding to Kepler’s and Ptolemy’s theories, or as Heilbron names them, equant theory, and pure eccentric or perspectival theory respectively.
Theory by 
Apogee distance 
Perigee distance 
Absidal Difference 
Kepler 


ae 
Ptolemy 
a(1 + e) 
a(1 – e) 
2ae 
Figure 26
In Figure 26, GH is of length 2s, OC refers to the separation between the Sun and the Earth at the apsides. Let GO = HO = d. Since triangles GOC and HOC are rightangled, by Pythagoras’ Theorem, d is equivalent to . Finally, b refers to half the apparent apsidal diameter of the Sun. Since b is small, it may be approximated by sinb. In the calculations that follow, the subscripts Π and A, when attached to a certain quantity, mark the same quantity as that measured at perigee and apogee respectively.
On the perspectival theory,
Since and s^{2 }is negligibly small,
By
similar derivation, and using
we obtain
The difference between the apsidal diameters is
and considering up to terms in e only,
where
Similarly, for the equant theory, by using and we get
According to Kepler, e = 0.036 whilst s = 30' .
Hence, if the perspectival theory holds true for the Sun, the difference in the apparent diameters of the Sun at the apsides would be
or if the equant theory holds true instead, the same quantity is
However, there was no consensus amongst the astronomers on which the correct value for the difference between the diameters at the apsides is. Cassini could not make any conclusion and thus turned towards using the Sun’s image along the meridiana at San Petronio. This piece of instrument offered higher precision and Cassini was able to measure the quality of interest, es by considering the length esh in the diameter of the Sun’s image along the meridiana, where h is the height of the gnomon.
Referring to Appendix C and Figure 26, a is the altitude of the Sun’s centre and b is half the Sun’s apparent size.
In triangle OPR, by Sine Rule,
Since we have Then,
For 2b small, Thus That is, the diameter PR of the central portion of the Sun’s image along the meridian line in Figure 23 is
Now, see Figure 27.
Figure 27
Due to the fact that and we obtain
Taking 2b as a small angle and s small such that s^{2} is negligibly small, the previous expression may be approximated by
sun’s diameter/sun’s distance.
On the equant theory,
On the perspectival theory,
The corresponding expressions for length PR in the two models are thus,
and .
Given the values of a at winter solstice WS, and summer solstice SS are 22˚ and 69˚ , we may calculate the difference ∆l between the diameters of the image at WS and SS which occur close to the apsides.
We note that ∆I = PR_{WS}  PR_{SS}
Therefore, in the equant theory,
Or,
In the perspectival theory,
which gives
Take σ = 30' , e = 0.036, and h=27.1 m. This gives
In order for a decisive confirmation of one theory over the other, Cassini had to measure she to within 8.5mm; otherwise, a finding of ∆l = σh(6 + 6e) would have decided nothing.
Later on, Cassini wanted to find a way to make measurements not only when the Sun is at perigee or apogee with respect to the Earth. He found that the relationship between the change in apparent diameter, ∆ρ of the Sun in any time interval, ∆t and the change in the Sun’s apparent position, ∆(θ – M) in the same time interval, is twice as big with a whole eccentricity as compared to a bisected one. Here, θ is the Sun’s angular distance from perigee, or the true anomaly; M is the displacement measured from the centre of the orbit C, or the mean anomaly. At time t, M is equivalent to wt or where T is the period of the Sun’s orbit.
Figure 28
Figure 28 illustrates Cassini’s method being applied on Ptolemy’s model. Applying Sine Rule to triangle CSE, we have
In addition, by applying Cosine Rule to triangle CSE, we get
Hence,
From the expression above, we obtain,
and considering up to terms in e only,
Therefore, by (1) and (2), we get
Since a is small, sina = a. That is,
Due to the fact that θ_{p} = M + α, we have ∆θ_{p} = ∆M + ∆α.
Thus, by differentiation and up to terms in e,
Recall that 2β = Sun’s diameter / Sun’s distance.
Therefore, the change in apparent diameter in ∆t is
Hence, in interval ∆t, the ratio of the ∆ in apparent diameter of sun to the ∆ in inequality in sun’s apparent position is
(1)
Figure 29
Figure 29 illustrates Cassini’s method being applied to Kepler’s model. Applying Sine Rule to triangle XSC, we have
In addition, with the same rule applied to triangle ESC, we have
where θ_{k} = M + β.
Consider β_{1} and β_{2} to be small angles then each of them may be approximated by sinβ_{1} and sinβ_{2 }respectively.
Since β = β_{1} + β_{2},
β is small and thus, by smallangle approximations,
By applying Sine Rule to triangle XES, we have
Hence, when b is small,
Similar to earlier working, since θ_{k} = M + β we have ∆θ_{k} = ∆M + ∆β
Thus,
by differentiation up to terms in e.
Therefore, the change in apparent diameter in ∆t is
(by differentiation)
Hence in interval ∆t, the ratio of the ∆ in apparent diameter of sun to the ∆ in inequality in sun’s apparent position is
(2)
Comparing (1) and (2), we observe that the latter is half the result of the former. Using the meridiana, Cassini made measurements that confirmed that the relationship between ∆(θ  M) and ∆ρ was indeed that expected for the theory of the bisected eccentricity. Thus, Kepler’s theory is confirmed as the correct model.