USE OF MERIDIANA

There had been a great controversy between adherents of Ptolemyís traditional solar theory and proponents of Keplerís ďbisection of the eccentricityĒ 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 Ptolemyís model, the solar eccentricity is of value e, then the same quantity would be of value in Keplerís model.

Figure 25

Referring to Figure 25(a), we see that at perigee, the distance between Sun and Earth is shorter on Ptolemyís model than that on Keplerís. From 25(b), the opposite occurs: at apogee, the same distance is longer on Ptolemyís model than that on Keplerís. 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 Sunís apparent diameter, which is inversely proportional to its separation from the Earth. Astronomers had found that the Sunís apparent diameter at mean distance ranges from 30' 30" to 32' 44". In addition, the difference between the Sunís 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 Keplerís and Ptolemyís 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 Ė 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 right-angled, by Pythagorasí 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 s2 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 Sunís 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 Sunís 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 Sunís centre and b is half the Sunís 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 Sunís 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 s2 is negligibly small, the previous expression may be approximated by

 sunís diameter/sunís 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 = PRWS - PRSS

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 Sunís apparent position, (θ Ė M) in the same time interval, is twice as big with a whole eccentricity as compared to a bisected one. Here, θ  is the Sunís 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 Sunís orbit.

Figure 28

Figure 28 illustrates Cassiniís method being applied on Ptolemyís 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 θp = M + α, we have ∆θp = ∆M + ∆α.

Thus, by differentiation and up to terms in e,

                           

Recall that 2β = Sunís diameter / Sunís 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 sunís apparent position is

                                    (1)

Figure 29

Figure 29 illustrates Cassiniís method being applied to Keplerís model. Applying Sine Rule to triangle XSC, we have

In addition, with the same rule applied to triangle ESC, we have

where θk = M + β.

Consider β1 and β2 to be small angles then each of them may be approximated by sinβ1 and sinβ2 respectively.

Since β = β1 + β2,

β is small and thus, by small-angle approximations,

                                    

                                                          

By applying Sine Rule to triangle XES, we have

Hence, when b is small,

Similar to earlier working, since θk = M + β  we have ∆θk = 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 sunís 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, Keplerís theory is confirmed as the correct model.