We ran the Dershowitz and Reingold's calendrica package to get values for the occurrence of a kshaya month. Since searching for a kshaya month is computationally very heavy [28], we used a table prepared by Saha and Lahiri (table 22 in the book) [29] as a starting point. We also tabulated results for non-kshaya months, specifically years with gaps of 19, 46, 65, 76, 122 and 141 years respectively.

It must be noted that all cases tabulated previously have been calculated according to Surya Siddhantic rules and that we may get a different set of results if calculated according to ephemeris calculations. Indeed, as Chatterjee has pointed out, there was a difference in 1964 CE; ephemeris calculations showed Margasira to be kshaya (and Karthika, Chaitra to be adhika), while as we've seen, Surya Siddhantic computation showed Pausa to be kshaya (and Asvina and Chaitra to be adhika) [30]. Chatterjee, however, seems to be in agreement with Dershowitz and Reingold in saying that there was a kshaya in Magha in 1983 CE [31], despite his use of ephemeris calculations.

What do we get from all this? We see that a kshaya month can occur every 19, 46, 65, 76, 122 or 141 years. Indeed, Saha and Lahiri's tabulation provide us with the following frequencies of occurrences for gaps between kshaya months:

Interval | Number of times occurring |

19 | 11 |

46 | 3 |

65 | 1 |

76 | 1 |

122 | 1 |

141 | 6 |

We therefore see that between 525 CE and 1985 CE, kshaya occurred 11 times with a gap of 19 years, thrice with a gap of 46 years, six times with a gap of 141 years, and once each with gaps of 65, 76 and 122 years. The obvious question one would like to ask would be why. Why does kshaya occur only in these gaps?

To answer this better, we re-iterate what causes kshaya in the first place. We already said that a kshaya would occur when two consecutive samkrantis occur between two Amavasyas. That is to say, when a solar month is shorter in length than, and is completely enclosed by, a (an amanta) lunar month. Saha and Lahiri go on to say that the “maximum duration of a lunar month exceeds the lengths of the solar months only in the case of Margasira, Pausa and Magha” [32] and that, therefore, kshaya is possible only in these months.

This would explain the solar month part, but what of lunar? How can the lunar month be bigger than the solar month? Ala'a Jawad has some answers; in his article, he suggests that the canonical synodic month, a lunar month between two consecutive phases of the Moon, is not constant in length. Indeed, he goes on to say that between 1600 and 2400 CE, the synodic month extends in length from 29 days 6 hours and 31 minutes to 29 days 19 hours and 59 minutes [33]. Moreover, he says that the “longest lunar months … occur when the date of the new Moon coincides with apogee” [34]. A brute-force search for the longest synodic month definitely won't give us a kshaya; for kshaya to occur, the lunar month needs to be only bigger than its solar counterpart and more importantly, completely encompass it. Indeed, Jawad says that the longest synodic month occurred in 1610 CE, a year which occurs within the 141 year long kshaya hiatus between 1540-1541 CE and 1680 – 81 CE.

We therefore search for other clues to unscramble kshaya. On a purely arithmetic perspective, we observe the following:

19 | = 19 * 1 |

46 | = 19 * 2 + 8 |

65 | = 19 * 3 + 8 |

76 | = 19 * 4 |

122 | = 19 * 6 + 8 |

141 | = 19 * 7 + 8 |

That is to say, the year-gaps are in the form 0, 8 mod 19.

Is it possible then, that the kshaya month has something to do with the Metonic cycle? The Metonic Cycle is a fairly well documented phenomenon; first observed by the Greek astronomer Meton, every 19 years, the lunar dates overlap with the tropical ones. The underlying mathematical reason is simple: 19 sidereal years contain 19*365.242189 = 6939.6 solar days, while 235 synodic months (with a mean of 29.53 solar days) contain 235*29.530588853 = 6939.68 solar days. The lengths overlap. But this obviously is neither necessary nor sufficient; it might be useful for the dates to repeat, but it definitely doesn't fulfil the requirement for kshaya.

One suggestion therefore, might be that the kshaya occurs when the number of solar days of a sidereal year is equal to that of a synodic month, which in turn is equal to that from an anomalistic month. An anomalistic month is defined to be the time – period between two consecutive perigee passages and has a mean value of 27.55455 days. Taking these average values, we calculate the average values of solar days in whole numbers of synodic and anomalistic months (canonical kshaya years shaded for reference):

Interval | Occurrence | Modulo | Solar year | Synodic months | Anomalistic months |

19 | 11 | 1*19 | 6939.601591 | 6939.68838 | 6943.7466 |

27 | 0 | 1*19+8 | 9861.539103 | 9863.216677 | 9864.5289 |

38 | 0 | 2*19 | 13879.20318 | 13879.37676 | 13887.4932 |

46 | 3 | 2*19+8 | 16801.14069 | 16802.90506 | 16808.2755 |

57 | 0 | 3*19 | 20818.80477 | 20819.06514 | 20831.2398 |

65 | 1 | 3*19+8 | 23740.74229 | 23742.59344 | 23752.0221 |

76 | 1 | 4*19 | 27758.40636 | 27758.75352 | 27774.9864 |

84 | 0 | 4*19+8 | 30680.34388 | 30682.28182 | 30695.7687 |

95 | 0 | 5*19 | 34698.00796 | 34698.4419 | 34718.733 |

103 | 0 | 5*19+8 | 37619.94547 | 37621.9702 | 37639.5153 |

114 | 0 | 6*19 | 41637.60955 | 41638.13028 | 41634.92505 |

122 | 1 | 6*19+8 | 44559.54706 | 44561.65858 | 44555.70735 |

133 | 0 | 7*19 | 48577.21114 | 48577.81866 | 48578.67165 |

141 | 6 | 7*19+8 | 51499.14865 | 51501.34696 | 51499.45395 |

Broadly speaking, we might summarize the above table as thus: for the most part, the number of solar days in solar years, synodic and anomalistic months overlap in kshaya years. However, this overlap doesn't occur only in kshaya years; as the table shows, there's an overlap for 133 years as well. Does this, then, explain the kshaya phenomenon? We might summarize it as being strongly suggestive, but definitely not conclusive.

- K.D. ABHAYANKAR,
*Our Debts to our Ancestors,*in "Treasures of Ancient Indian Astronomy" (ed. K.D. Abhayankar and B.G. Sidharth), Ajanta Publications, Delhi. 1993. - Helmer ASLAKSEN,
*The Mathematics of the Chinese Calendar,*preprint, National University of Singapore, 1999. - Apurba Kumar CHAKRAVARTY and S.K. CHATTERJEE,
*Indian Calendar from Post-Vedic Period to AD 1900,*in "History of Astronomy in India" (ed. S.N. Sen and K.S. Shukla), Indian National Science Academy, New Delhi, 1985. - S.K. CCHATTERJEE,
*Indian Calendric System,*Publications Division, Ministry of Information and Broadcasting, Government of India, 1988. - Nachum DERSHOWITZ and Edward M. REINGOLD,
*Calendrical Calculations,*Cambridge University Press, 2001. - Nachum DERSHOWITZ and Edward M. REINGOLD, Calendar Tabulations – 1900 to 2200, Cambridge University Press, 2002.
- Nachum DERSHOWITZ and Edward M. REINGOLD,
*Indian Calendrical Calculations.* - Ala'a H. JAWAD,
*How Long is a Lunar Month?,*in Sky and Telescope, November 1993. - Akshay REGULAGEDDA,
*Panchanga-Tantra: The Magic of the Indian Calendar System,*Undergraduate Research Opportunities Programme in Science (UROPS) thesis, National University of Singapore, 2002. - M.H. SAHA and N.C. LAHIRI,
*History of the Calendar in Different Countries Through the Ages (Part C of the Report of the Calendar Reform Committee),*Council of Scientific and Industrial Research, New Delhi, 1992. - Robert SEWELL and Sankara Balakrishna DIKSHIT,
*The Indian Calendar,*Motilal Banarsidass Publishers Pte. Ltd., 1995.

- [28] DERSHOWITZ and REINGOLD (2002), p. 24.
- [29]SAHA and N.C. LAHIRI, p. 250.
- [30] CHATTERJEE, p. 38.
- [31] DERSHOWITZ and REINGOLD (2001), p. 269.
- [32] SAHA and N.C. LAHIRI, p. 250.
- [33] JAWAD, p. 76.
- [34] Ibid.

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Helmer Aslaksen

Department of Mathematics

National University of Singapore

helmer.aslaksen@gmail.com

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