It’s now known and proved that the planetary orbits around the Sun are elliptical in nature. This is the first one of the famous three Kepler laws. It is widely accepted that nobody knew about this fact before the early 17th century. But is it so? Did nobody before Kepler notice such variations?
The second chapter of Surya Siddhanta is dedicated towards calculating true values of planets. In this chapter, we learn that terminology used to find the true planets is to apply manda phala and sheeghra phala corrections (which are equation of apsis (or center) and equation of conjunction respectively) on the mean planet positions. Two equations, the manda phala (manda correction) and sheeghra phala (sheeghra correction), are calculated based on the manda/ sheeghra paridhis (circumferences) and manda/ sheeghra kendras (centers) as referred from mandocha (the apsis)/ sheeghrocha (the reference for conjunction) respectively. The chapter specifies the manda paridhis for all the planets; and sheeghra paridhis for each of the panchtara planets (namely Mars, Jupiter, Saturn, Mercury and Venus). The concept seems somewhat analogous to the epicycle concept that Greek astronomers such as Ptolemy also mentioned. But though some critics alleged that the siddhantic theories are in ditto lifted from Greeks, there are many differences. One such difference is varying the epicycle size i.e. specifying different paridhis (circumferences) for even and odd quadrants for both manda and sheeghra phalas.
This is a very important question. Any true seeker would definitely think about an elliptical motion while looking for the probable answer. This is what Burgess also did and mentioned it briefly in his commentary on the surya siddhanta. His comments can be summarized as under:
1. It leads us to a quasi ellipse, a rather ovoid shape, a shape that is flattened in the 1st/4th quadrant and bulged in the 3rd/2nd quadrant.
2. In case of sheeghra, the axis of such shape would be pointed towards Earth.
3. It can be related to the virtual orbit theory that Ptolemy mentioned where the center of an eccentric circle lies at half of the radius of the epicycles.
And then the final comment that the solution to this difficulty (if at all is provable) should be put for further investigations; while raising doubts that the phenomenon could be linked to empirical reasons.
None of the other commentators of Surya Siddhanta have touched this subject or tried to explain these variations.
Regarding point 3 above, any person, who has understood the eccentric concept even up to basic level (that Burgess has also detailed himself to great extent in his commentary work), can easily write that “there is no linkage” and the argument is invalid. If someone tries to link the reason for phalardha (half the correction) in case of panchtara (outer and inner planets) planets in the surya suddhanta to the mentioned phenomenon of Ptolemy, still one could digest, but how and where it is linked to variations in the manda and sheeghra paridhis (circumferences of epicycles). Two concepts are a way apart. I would also mention here that the manda paridhis are varied for Sun and Moon too (not just the “panchtara” planets) which was probably missed by Burgess while presenting his above said comment. Hence the linkage between the two sounds illogical.
Regarding doubt raised in the final comment, there is no point in assuming that ancients were not keen on correcting minutely with respect to the observations without any valid reason. Maybe it was due to accuracy errors or wrong understanding of 54 seconds precession rate as reckoned from the mentioned ayana calana verses or anything else that might have contributed him to comment like this (Note: the ayana concept has been summarized in the articles under “Nirayana” section on this website and is covered in detail in my book “Taming the Untamed”), but we have no strong reasons to doubt that they were not interested or didn’t attempt at all. There were deep observations and the reason why they specified varying manda/ sheeghra circumferences too to align the calculations to those observations.
Let us now discuss the other two comments, number (1) and (2), since these two are logical and valid arguments.
If we try to apply the variations in the circumferences on the various geometries that are presented by Burgess and other commentators, we find that when we apply the variation in the first quadrant, we move the center of the eccentric circle to the new position which aligns to the possible ellipse center but as soon as we move to the second quadrant, the center returns to the original position and we are left with an ellipse that has its semi major axis as R+delta (“delta” being the component due to variation in the circumferences) in the first quadrant but the semi major axis now becomes semi minor axis of the other half of the ellipse with value R-delta. It is like we have one half of an ellipse that has the semi major axis equaling R+ delta and the other half of another ellipse that has the semi minor axis as R-delta. This means it leads us to what Burgess pointed as “ovoid shape”. This also means that despite this variation, the overall major axis of this pseudo or quasi ellipse remains equal to 2R (R+delta+R-delta), the diameter of the eccentric circle itself.
If we browse through the commentaries of siddhantic works other than surya siddhanta, few authors have tried to explain these variations linking it to an ellipse. The attempts are genuinely commendable, though unfortunately they have used the same geometry to explain the ellipse part that leads us to a quasi-ellipse only as indicated by Burgess.
I would term the variations in the circumferences (different paridhis at the end of even and odd quadrants) as mentioned in the surya siddhanta are definitely pointing to the elliptical motion of the planets.” The circumferences were varied since the observed orbits were not found exact according to the eccentric circle. I would claim that the fundamental mentioned in Surya Siddhanta is in line with the fundamental of an ellipse, a true ellipse. Though they did not mention it as an elliptical motion or defined an ellipse, they discovered the variations and put them as varying circumferences, long before Kepler. Of course, there are shortcomings which can obviously be expected in a non-computer age.
This space is bit short and not the right platform to explain my findings in detail (I would definitely write a separate paper or so in near future), but here are my inputs until the date:
1. The geometry specified in the commentary works cannot explain this elliptical concept nicely.
2. One can’t achieve the ellipse by just applying the “sine” formula (as specified in the surya siddhanta to find the instant circumference of the phala cycle) in each and every case.
3. The variations in circumferences are not microscopically exact to represent the ellipse vis-à-vis the focus specified.
4. The ellipse formed owing to variations in the sheeghra phala circumferences is still a concern. It means argument number 2 as raised by Burgess is valid. The choice of using true Sun instead is a good candidate. Maybe by the time I write my inputs in detail, I hope to have enough to comment exhaustively on this too.
March 16, 2016
Devinder Dhingra
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