Intercalary months & consistency of a theory: An Illustration

What we do know:

(1) Mahabharata calendar is Luni-solar.

(2) Adhika masa (intercalary months) were employed to synchronize Lunar year (~354 days) with solar year (~365 days).

(3) Lunar months were named, very much like in our times, e.g. Chaitra, Kartika, Margashirsha, Magha, etc.

(4) Researchers have made cases in favor of both (a) Amanta (beginning and ending with Amawasya) and (b) Purnamanta (beginning and ending with Purnima) system of lunar months

(5) Specific references to Adhika masa are scant within the Mahabharata text.

(6) There are two references to Adhika masa from the Mahabharata text and are shown below:

 (1) विराट पर्व  (CE ४७:३-४, GP ५२:३-४)
तेषां कालातिरेकेण ज्योतिषां च व्यतिक्रमात
पंचमे पंचमे वर्षे द्वौ मासावुपजायत:
तेषामभ्यधिका  मासा: पंच द्वादश च क्षपा:
त्रयोदशानां  वर्षाणामिती  मे वर्तते मति:
 (2) विराट  पर्व  (Bombay edition ४७:३)
गते वर्षद्वये चैव पंचपक्षे  दिनद्वये
दिवसाष्टमे  भागे  पतत्येकोधिकमास:
What we do not know:
(1) What specific method was employed (during Mahabharata times) to determine timing for the insertion of ‘adhika masa’ (intercalary month).
Against this background, let’s look at two interpretations of Bhishma’s words
(1) Adhika masa was inserted one at a time and amounting to approximately (and only approximately) about 2 adhika masa per every 5 years (on an average)
 Let's call this 'Oak' interpretation.
(2) Two adhika masa(s) were inserted at a time in a consecutive fashion.
 Let's call this 'Phadnis' interpretation.
Shri Phadnis also interprets this to mean 2 consecutive adhika masa(s) were inserted after 58 Lunar months. Shri Phadnis also states that ‘2 consecutive months after 58 lunar months is more appropriate than 2 consecutive months after 60 lunar months.
We will ignore the discussion of 58 vs. 60, for brevity, since my specific objection is to his interpretation of ‘2 consecutive adhika masa(s)‘.
Shri Phadnis claims that “2 consecutive adhika masa(s)” is the correct translation of this Mahabharata  text, specifically (पंचमे पंचमे वर्षे द्वौ मासावुपजायत:).
He accepts translation of second shlok as referring to Bhishma’s calculation of about 5 months and 12 nights of Adhika masa correction corresponding to 13 years of exile (of Pandava brothers).
 विराट पर्व  (CE ४७:३-४, GP ५२:३-४)
तेषां कालातिरेकेण ज्योतिषां च व्यतिक्रमात
पंचमे पंचमे वर्षे द्वौ मासावुपजायत:
तेषामभ्यधिका  मासा: पंच द्वादश च क्षपा:
त्रयोदशानां  वर्षाणामिती  मे वर्तते मति:
Consistency of a theory: An illustration
One of the key aspect in determining the worth of any theory is its consistency.
Consistency is tested by multiple means:
(1) logical inferences due to theory
(2) contradictions with background assumptions of a theory
(3) empirical observations conflicting with interpretations/predicted consequences of a theory
(4) Ability of a theory to explain/corroborate all ‘relevant’ observations for which theory makes predictions
(5) Internal contradictions of a theory with the evidence it is trying to explain
Against this background, let's evaluate comments of Shri Phadnis.
He writes...
“Shri. Oak cannot translate what Bhishma said as he likes. पंचमे पंचमे वर्षे मासद्वयम must be translated according to sanskrit grammer rules only. It translates as ‘a pair of months in every fifth year’. Whether, EXACT or sufficiently accurate, that was the system in Bhishma’s days.
About Virata Parva Shloka from Bombay edition, it is a ‘mathematical formula’ and not a working system. It may be accurate. Can a new lunar month begin after end of the stated period? It is clearly an interpolation.”
My Response…
(1) Shri Phadnis is on the right footing when he asks “Can a new lunar month begin after end of the stated period?” in referring to inference of Mahabharata text from ‘Bombay edition’

विराट  पर्व  (Bombay edition ४७:३)

गते वर्षद्वये चैव पंचपक्षे  दिनद्वये
दिवसाष्टमे  भागे  पतत्येकोधिकमास:
(2) He is also correct when he states, referring to the same shlok, “it is a ‘mathematical formula’ and not a working system
(3) Based on (1) and (2) , Shri Phadnis states, “It is clearly an interpolation”.
Well, we do not know that (It is clearly an interpolation) for sure.  However, the fact that only Bombay edition has it (and numerous other editions do not have it) may allow us to make a case for treating this shlok as ‘plausible’ interpolation.
Fortunately, we do not have to resolve the issue of ‘interpolation’.
This is because the very argument Shri Phadnis employs in criticizing ‘Bombay edition sholk’ also applies to his own interpretation.
I assert that the remaining reference (other than Bombay edition shlok) for Adhika masa is also a mathematical formula and not a working system.
Let’s assume Shri Phadnis interpretation of taking “2 consecutive Adhika masa(s)” is the valid and correct translation and let’s try to make sense of this interpretation for Bhishma’s calculation of 5 months and 12 nights of Adhika masa corresponding to 13 years of exile.
  • Since 2 adhika masa(s) are inserted in a consecutive fashion, we can have Only 4 or 6 adhika masa(s) during any given period of 13 years.
  • For example, Let’s imagine that time of Pandava brothers beginning their exile coincided with the first day of Adhika masa – we will have 2 Adhika masa followed by 58 months (per Shri Phadnis formula) followed by 2 Adhika masa followed by additional 58 months.  At  this points 10 years of exile are over and we have 4 Adhika masa(s). We are also ready to get into 2 additional adhika masa(s).  Additional 36 months (corresponding to remaining 3 of 13 years of exile) would follow.  This would amount to having 6 Adhika masa(s).
  •  Now let’s consider other extreme scenario.  Let’s imagine that time of Pandava brothers beginning their exile coincided with end of the time of ‘2 consecutive Adhika masa’ and thus first set of adhika masa would occur only after Pandava borthers had spent 58 lunar months in exile.  This is when we have 2 adhika masa(s) followed by 58 months and again 2 adhika masa(s).  At this point10 years of exile are over and we have 4 Adhika masa(s).  The remaining time of ~36 months (3 years) begin, at the end of which exile would be over with no additional adhika masa.  This would amount to having 4 adhik masa(s)
None of these two limiting cases corroborate Bhishma’s calculation of 5 months and 12 nights of adhik masa.
So, is it at all possible to have exact 5 months and 12 nights of adhika masa correction corresponding to 13 years of exile and still reconcile this reference of Bhishma with interpretation of ‘2 consecutive Adhik masa’?
In theory, it is possible.
Visually we may show it something like this, beginning with the end (i.e. end of exile) and going backwards in times….
1 month and 12 nights of Adhika masa correction (in progress) when exile ended - going backwards 58 lunar months - 2 adhika masa - 58 lunar months - 2 adhik masa - balance of time enough to make total duration of 13 years of exile
 —
We evaluated three specific scenarios in the context of Shri Phadnis claim of ‘2 consecutive adhik masa’ and Bhishma’s calculation of 5 months and 12 nights corresponding to 13 years of exile.
Two scenarios either provide us with 4 or 6 months of Adhik masa correction.  The remaining third scenario, in principle, would provide us with exact correction of 5 month and 12 nights of Bhishma.  Voila!
 Except there is a small problem, or series of big problems.
  • Small problem is that this would mean Bhishma (and thus timekeepers in Mahabharata times) exactly knew when the exile began, appropriate timings of adhika masa insertions  and thus there should have been no confusion whatsoever whether Pandava brothes completed their 13 years of exile successfully or not.
  • If we are not willing to accept this inference, then we must admit that what Bhishma is referring to in two shlok is also a mathematical formula (and that too in a ‘back of the envelope’ calculation) and thus not a working system.
And don’t forget, this is the very reason why Shri Phadnis termed ‘Bombay edition shlok’ as ‘It is clearly an interpolation’.
To make this worse, these are not the only problems with ‘2 consecutive adhik masa’ interpretation.

Bigger problems include
  1. Comprehending practical rationale for synchronization of Luni-solar calendar, viz. occurrence of seasons (which are based on solar calendar) occurring with predicable/reasonable accuracy within a range of lunar calendar month/tithi.  Method of ‘2 consecutive Adhika masa’ would make system ‘out of sync’ for unduly extended durations and thus making the very act of synchronizing redundant.
  2. Beauty of Lunar calendar is that it enables an ordinary observer to test its accuracy via ‘Drik-Pratyay’ (visual observation of the position and phase of the moon).  This makes it a self-correcting system. Waiting for ~5 years to correct the system (i.e. synchronize) defeats the very reason for which the system was designed.  Otherwise ~purely solar calendar (Gregorian/Julian) or purely lunar calendar (Islamic calendar) are in use and they do the job they were designed to do.
  3. While we do not know the specific method employed in Mahabharata times for insertion of Adhika masa, what we do know is that method of inserting ‘2 consecutive adhika masa’ after 58 lunar months’ does not do anything to synchronize Luni-solar calendar.  In fact all it does is complete 5 year long (12 x 5 = 60) Lunar calendar (60 lunations).
  4. I may mention that even insertion of ‘2 consecutive adhika masa’ after every 60 lunar months would not solve the problem either.  Explaining this would require additional explanation.  I may attempt explaining it another time.  For now, curious may do their own calculations.
 
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