Bhishma’s words on Adhika masa – Plausible interpretations

I had quoted two references from the Mahabharata text, in the context of Adhik masa.

(1) Virata Parva (Bombay Edition 47.3)

गते वर्षदवये चैव पंचपक्षे दिनद्वये
दिवसाष्टमे भागे पतत्येकोधिकमास:

(2) GP Edition,Virata Parva: 52:3-4

तेषां कालातिरेकेण ज्योतीषां च व्यतिक्रमात
पंचमे पंचमे वर्षे द्वौ मासावुपजायत:

एषामभ्यधिका मासा: पंच च द्वादश क्षपा:
त्रयोदशानां वर्षाणामिति मे वर्तते मति:

One of the readers of my book insist on a specific interpretation as the only interpretation of the following Shloka (Mahabharata text). He considers first reference (Bombay edition) as interpolated. We will revisit the broader issue of ‘interpolation’ and problem associated with it, especially tendency of researchers to discard something ‘easily’ as interpolation, when it does not suit their argument.

Our reader is comfortable with the second reference (GP Edition -Virata parva 52:3-4) and uses this reference in support of his interpretation that two adhika masa were taken consecutively at the end of continuous time counting of 58 lunar (or solar?) months.

His specific point is that ‘two adhika masa’ (intercalary months) were taken in a consecutive fashion (adjacent to each other) during Mahabharata times and that they were NOT taken spaced at an interval (about ~2.5 to 2.8 years spaced out) as ‘assumed‘ by yours truly.

Against this background, let’s explore all plausible interpreatations, per claim of our reader, that can be derived from this second reference.

(2) GP Edition,Virata Parva: 52:3

तेषां कालातिरेकेण ज्योतीषां च व्यतिक्रमात
पंचमे पंचमे वर्षे द्वौ मासावुपजायत:

I translate it as —“two adhika masa (intercalary months) are taken every 5 years”.

Our reader translates them to mean ‘Two adhika masa (intercalary months) are taken consecutively/adjacent/next to each other, after a period of 58 months.

Our reader does not specificy the nature of these ’58 months’ (i.e., either solar or lunar in nature). We will analyze them by assuming both scenarios.

GP Edition,Virata Parva: 52:4

एषामभ्यधिका मासा: पंच च द्वादश क्षपा:
त्रयोदशानां वर्षाणामिति मे वर्तते मति:

I think (our reader would have to tell me otherwise) that there is no disagreement on translation in this case.

The translation goes something like this…

A period of 13 years based on solar year computation would lead to additional 5 months and 12 nights (of lunar months) which would have been (or to be inserted) inserted as ‘adhika masa’, in order to synchronize Luni-Solar Calendar.

While our reader and I differ on interpretation of first shlok (GP edition Virata Parva 52:3), I consider this second shlok (GP edition, Virata parva 52:4) decisive in reaching the correct interpretation.

Synchronizing Luni-Solar Calendar: Quick guide

Length of Solar year is about 365.25 days

(Actual length is slightly shorter than 365.25 (365.2422), but is ignored as not relevant for the issue being discussed here. However, it is of immense importance as it relates to determination of ancient Indian events)

Length of Lunar year (12 lunations) is about 354 days (354.36 days)

Thus at the end of each solar year, lunar tithi would recede by (365.25 – 354 = 11.25) days. The actual number is close to 10.88 days. In short, lunar tithi for a given day on solar calendar would recede by ~11 days at the end of one year.

In about 3 solar years the total gap would accumulated to ~ 11 x 3 – 33 days.

Thus, in our times, a lunar Adhika masa (intercalary month) is added on an average every ~2.8 solar years to adjust for this accumulated gap.

Days in 5 solar Years = 365.25 X 5 = 1826.25
Days in 5 lunar years (1 lunar year = 12 lunations) = 29.53 x 12 x 5 = 1771.8 days.

This leaves a gap of 1826.25-1771.8 = 54.45 days – 1.84388 lunar months.

Thus a correction of two lunar months (taken consecutively) would actually correct for gap of 1.84388 months, but will also generate an error, since correction would be of 2 months and NOT 1.84388 months. The actual magnitude of error (in days) would depend on the placement of these two consecutive adhika masa, during any 5 year solar cycle.

Let’s us begin with interpretation of our reader, and various plausible scenarios it generates…

If 58 months that he refers to, after which two consecutive months are inserted, are Solar months in nature, then the onus is on our reader to explain us what ‘lengths’ he is assuming for each of these 58 solar months.

This is because, considering our formula above, a gap (between Lunar and solar year) would accumulate to around 11 x 5 = 55 days, at the end of 5 solar years.

Unless we know the total time (and total duration of days due to 58 solar months) duration due to 58 solar months, we can not determine if this method of taking two consecutive Adhiaka masa can do the intended job.

The information supplied by our reader for his proposal (assuming 58 months he referred to were solar months) is insufficient to determine the usefulness of his proposal.

If 58 months that our reader refers to, after which two consecutive months are inserted, are lunar in nature, then this would amount to total days = 58 x 29.53 = 1712.74 ~ 1713 days.

Days in 5 solar Years = 365.25 X 5 = 1826.25
Days in 5 lunar years (1 lunar year = 12 lunations) = 29.53 x 12 x 5 = 1771.8 days.

This leaves a gap of 1826.25-1771.8 = 54.45 days – 1.84388 lunar months.

On the other hand, per interpretation of our reader, the actual gap would amount to 1826.25- 1712.74 = 113.51 days which amount to 3.84388 lunar months.

Thus a correction of two lunar months (taken consecutively), at the end of 58 lunar months would actually lead to ONLY partial correction (2 out of total desired 3.84388 lunar months) and leave a gap of 1.84388 lunar months.. or that of about ~55 days.

Interestingly, Bhishma refers to accumulation of 5 months and 12 nights in ‘adhika masa’ calculations for the period of 13 years.

This very fact states that Bhishma is assuming 13 solar years and our reader would (should) have no problem with this conclusion (13 being solar years in Bhishma’s mind).

Thus I wrote….

If the claim of this seeker (भीष्माने ५८ महिन्यांनंतर एकत्र दोन अधिक महिने घ्यायचे असतात असे स्पष्टपणे म्हटलेले असताना) is true, then Bhishma should not have stated extra 5 months, at the end of 13th year!

Since, according to this seeker, Bhishma would not refer to additional-Adhika-Intercalary 5th or 6th month until the end of 15th year. This is because per the claim of our seeker, next set of 2 adhika masa would not be added until the end of Year 15!

Full article can be read here…

Our reader responded to my above comment of ‘2 adhika masa at the end of 15th year, and not 13th’

“Why the third pair of adhika masas has to come at the end of 15 years? The first pair of adhikamasas could have occured even one year of vanavas and third pair in 11th year. In 12 years of vanavas and 1 year of adnyatavas, either 4 extra months or 6 extra months could have ocurred. If only four extra had occured, pandavas would have spent less than 13 solar years, 156 + 4 lunar months. This would have suited them!”

Let’s see if solution supplied by our reader solves the problem raised by me (i.e. insertion of any adhika masa beyond 4 months – corresponding to first 10 years, should have waited until the end of 15th year)

(3) – A

Let’s take a best possible scenario, as would be envisioned by our reader, in defense of his proposal (58 months followed by 2 consecutive adhika masa)

We will imagine (to give best shot for success to this proposal of our reader – 58+2) that when Pandava’s began Vanavas indeed was the beginning of first Adhika masa.

Thus Pandava begin Vanavas – 2 adhika masa

followed by 58 months + 2 adhika masa
—completion of just over 5 years and we already have 4 adhika masa.

Additional 58 months go by, and then 2 additional adhika masa…

At this point, we have just over 10 years (2 + 58+2+58+2) and already 6 Adhika masa.

Essentially, this is the scenario our reader has painted for us, but then he does not explain why Bhishma refers to adhika masa need of ONLY 5 months and 12 days (and not 6 Adhika masa)

An hypothesis/interpretation of 2 consecutive Adhika masa after 58 months do not synchronize Luni-Solar calendar, generates additional error of odd days and the hypothesis/interpretation does not bother to suggest a solution to account for this additional error. Over time, error would only get worse.

(3) – B

Alternate interpretation of Bhishma can be understood in the following fashion.

Bhishma is making only a crude calculation…

Virata parva 52:3 states that 2 Adhika masa occur in 5 year time period/interval.

And thus, mathematically,

If 5 year corresponds to 2 Adhika masa

13 years corresponds to ?

Solving, (13/5) x 2 = 5.2 months. Taking a month duration of ~30 days: 0.2 month = (30 x 0.2) = 6 days.

So correct answer is rather 5 months and 6 days and not 5 months and 12 nights (approximate enough). BTW, I am not suggesting that one accept this approximation, but then onus is on that person objecting to this approximation, to explain why Bhishma said what he(Bhishma) said.

Per this interpretation, Bhishma is simply illustrating a rough math (back of the envelope calculations), then that is great and that may or may not be enough to pacify Duryodhana, however, this interpretation nowhere leads to the claim of our reader that Bhishma is referring to tradition/method/process of taking 2 consecutive adhika masa after a time interval of 58 months!


(1) A method (as claimed by our reader for the Mahabharata times) of 58 months followed by 2 consecutive adhika masa is INCAPBLE of synchronizing Luni-Solar Calendar. At the end of first 5 year period, an error of few days would occur. This error would depend on assumptions for the solar or lunar nature of 58 months. In first case of solar 58 months, we do not have sufficient information to calculate the error. In case of 58 lunar months, the error is huge (amounts to about ~55 days) and our reader has not suggested a solution to correct for this. Of course, any solution would immediately disqualify the very claim of our reader for the method of 58+2.

(2) Our reader also claims that Virat Parva reference 52:3-4 refers to the method of 58 + 2, or at a minimum, two consecutive adhika masa. I have shown with the calculations that such a method is incapable of synchronizing the Luni-Solar Calendar. In addition, second part of this very reference shows that interpretation of our reader (58+2, 4 adhika masa or 6 adhika masa) does not agree with Bhishma’s calculation of adhika masa correction of 5 months and 12 nights, even after giving every possible benefit of doubt (for example, location for insertion of Adhika masa – for the Vanavas period of Pandavas).


4 thoughts on “Bhishma’s words on Adhika masa – Plausible interpretations

  1. 1. Solar year of 365.25 days and Lunar 12 month year of 354.36 days cannot be reconciled accurately by any simple arrangement of extra lunar months.
    2. Bhishma has said पंचमे पंचमे वर्षे … This can be translated only as ‘in the fifth year’ and not as ‘in five years’.
    3. Taking one month after 29 months or 2 months after 58 months will give same matching or mismatching.
    4. The probability of six adhikamasas occuring within 13 Lunar years i.e. 156 lunar months is in fact MORE than only 4 adhikamasasa. 2+58+2+58+2 =122 leaves 34 months balance which can divided between before and after as 0+34, 1+33 etc to 34+0, in 35 ways. On the other hand, 58+2+58+2 = 120 leaves 36 months balance to be added at end. Opening 58 can be reduced by 1 and 36 will be 37. Reduce opening by 23 and you get 59 months calling for 2 extra at end of 58th. So only 22 acceptable ways. A probability ratio of 35:22. Shri. Oak’s remarks on this point are unwarranted and made in haste, to say the least!
    5. The Jews use Lunar calender and compensate by taking 1 adhikamasa (Always the first month in the year) in 7 specific years in a 19 year cycle. 19×354.36 + 7x 29.53 and 19×365.25 match as close as 6940.8 and 6939.75 days. They claim it is in vogue from the beginning of their ‘shaka’.
    6. I do not know whether even our refined system now in vogue of taking adhikamasa when no rashi transit of Sun occurs in it and a kshayas masa when two rashi transits occur in a lunar month, gets a closer match and if so, in how many years’ cycle.
    7. Bhishma no doubt calculated Pandawas’ liability of extra period in 13 years to match 13 solar years and, in effect said that Pandawas have completed 13 solar years by spending 5 months and 12 nights extra. Mahabharata is silent about the month and tithi of the day when dyoot was played. It mentions only krishna ashtami as the tithi but no month when Arjun appeared for fighting with Kauravas. So one cant say with any degree of assurance whether they had reached the 13th iteration of the dyoota tithi/month or if not, how many days they fell short. But fell short they did. Duryodhana repeatedly said Pandawas have not completed 13 years, because he obviously knew the tithi/month of dyoot day and that of the day when Arjun came out. If Pandawas counted 5 months and 11 days instead of 6 months, they fell short by 18-19 days.
    8. Two extra months per 58 regular months leads to (156 / 58) x 2 = 5 months and 11 days.
    9. Shri. Oak should explain how the wording of the Bombay edition shloka is to be translated and how it could have been implemented in Mahabharat days or even now for that matter, for determining which particular month was an adhikamasa and which was not. Interval of 2 years, 5 months, 2 days and 1/8th day between two adhikamasas (as I translate it) may be mathematically correct but no way you can get adhakamasas separated by this period in reality. You need a workable formula.
    10 I do not know what formula for designating a particular month as adhika was in vogue prior to the introduction of our current system based on Rashisankramana of the Sun and since when. If Shri. Oak knows it he should state it and also since when it has been in vogue. In mahabharata days, I am sure it was just 2 extra after 58 regular, simple and sufficiently accurate. Lunar year of 354 days extends by a factor of 60/58 over several years to average 366 days. With a correction unit of one lunar month at a time, a closer match cannot be obtained by a simple and workable formula.

  2. I did some arithmetic as a matter of curiosity. 58 months followed by 2 extra months actually overcompensates the difference between 365.25 and 354.36 days of solar and lunar years. Over 29 years, solar years will cover 10592.2 days. 29 groups of 12 lunar months will cover 10276.4 days. But if you add the six pairs of adhikamasa you get 29.53 x12 more days which are 354.4 days. These add up to 10630.3 days! Even if you leave out the 12th adhikamasa, the total is still 10601.3 days, 9 days more than 10592.
    I find that over a period of 87 solar years, instead of 36 adhikamasas arising out of 58 +2 formula, (or 29 +1 for that matter),if you take only 32, (which four to omit? I have no view or formula.) one ends up with a difference of only 2.4 days! I wonder whether we ever had any such method.
    So what was our method prior to the current method and how close a match it provided? Since when is it in vogue?
    One wonders how the Jews hit upon 7 extra in 19 years which gives a match with only 1 day difference! hats off to them.

  3. Shri. Oak does not realize that with a full lunar month as a correction unit, matching of Lunar year and solar year can never be perfect. It is matter of great wonder that the Jews have a particular cycle of 19 years in which 7 extra months are taken and the match is almost perfect. If the date of first day of Lunar calendar for the last large number of years is checked, very rarely, if at all, the same date will occur.
    If 2 extra months are taken after 58 months it amounts to increasing the average length of lunar month by 1/29. Since a lunar month is of 29.53 days, 1/29th of that is 1.01 day. The average length becomes 29.53 +1.01 = 30.54 days. Solar year of 365.25 days gives the length of a solar month of 30.44 days. This was surely a sufficient match for mahabharat days. I have never claimed that Bhishma’s formula was extremely accurate.
    2 extra months after 58 months gives a better match than 2 extra months after 60 months.
    In any case 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.

  4. I am not sure what specific point Shri Phadnis is trying to make (or trying to refute) with his insistence on 2 consecutive months.

    (1) I have shown the huge error introduced by 2 consecutive months after 58 months. I will elaborate it some other time
    (2) The objection to Shri Phadnis translation is not because of non-Exact correction. That would be the case anyways. However, there are serious issues with the consequences (due to his interpretation of 2 consecutive months as adhika masa). More on that later.
    (3) Shlok from Bombay edition is a ‘mathematical formula’ (and since it exists only in Bombay edition, that can be treated as outlier and thus plausible interpolated/inserted). However, by the same logic of mathematical formula, the statement of Bhishma amounting to 5 extra lunar months (adhika masa) and 12 nights is also a mathematical forumula. And, lo and behold, based on his previous statement of, per Shri Phadnis translation of 2 consecutive months of adhika masa.
    (4) 5 months and 12 nights extra (due to Adhika masa correction) over a period of 13 years CAN NOT be reconciled with interpretation of ‘2 consecutive adhika masa’ during a continuous period of 13 years.. no matter when the Adhika masa(s) were inserted.

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