[INDOLOGY] Tithi & synodic month

Bill Mak bill.m.mak at gmail.com
Thu Nov 5 22:35:06 UTC 2015

Dear Artur,

Sphujidhvaja in Yavanajātaka 79.5 gives the definition of a tithi as 63/64 of a civil day:

dinaṃ catuḥṣaṣṭilavonam āhus tithiṃ dyuśabdākhyam ahas tu sarvam |

triṣaṣṭibhāgena yutaṃ sahasraṃ yuge ’vamānām apasaptaṣaṭkam ||5||

They say that a day (dinam) minus 1/64 [of a day] is a tithi; on the other hand, a day (ahar), called by the word dyu, is one whole [tithi] plus 1/63 [of a tithi]. The number of omitted tithis (avama) in a yuga is equal to 1000 minus "7 times 6" (i.e. 1000 - 42 = 958).

Pingree's translation is somewhat different and but I believe his maths is incorrect since t = 1- 1/64 d, d = t + 1/63. His reading of cd was based on his own emendation and not on the ms (or even his own reading in the apparatus!):

They say that a tithi equals a day minus 1/64th, but that every day equals a tithi plus 1/60th. In a yuga there are 990 seasons (ṛtu), (each) consisting of 62 (tithis).

63/64 is used only as an approximate value, since the algorithm for the "caturmānavidhāna" (79.6 ff.) yields the value 60272/61230 which would give a better synodic month of 29.5306222... (given also in 79.12) compared to 29.53125 (cmp. modern value: 29.53059...).

Hayashi Takao pointed out to me that the expression for 64/63 can be easily derived from 61230/60272 through a popular method of approximation by reducing a series of fraction. I would be interested in other instances of 64/63, 1/64 or 1/63. If you know any, please let me know.

For further explanations and notes, see my critical edition:
Mak, Bill M. 2013. "The Last Chapter of Sphujidhvaja's Yavanajātaka critically edited with notes." SCIAMVS 14: 59-148. In particular, pp.78-80, 90-91.

Best regards,


Dr. Bill M. Mak
Hakubi Associate Professor of History of Science

Institute for Research in Humanities, Kyoto University
Yoshidahonmachi, Sakyo-ku, Kyoto, Japan 606-8501
〒606-8501 京都市左京区吉田本町

email: mak at zinbun.kyoto-u.ac.jp

copies of my publications may be found at:

On 2015/11/06, at 6:47, Artur Karp wrote:

> Dear Colleagues,
> Is there any evidence for the use of the fraction 63/64  – to determine the length of the lunar day (tithi)? In ancient Babylonian-Greek astronomy? In ancient Indian astronomy?
> The result of the division, if multiplied by 30, would give the length of the synodic month  – as 29. 53125d.
> Your help in the matter would be greatly appreciated,
> Artur Karp
> University of Warsaw, South Asian Studies Dept., Emeritus
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