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Bad calculations: another missing link in the history of the rabbinic calendar

Sacha Stern

The Jewish calendar that is in almost universal use today, and in the Middle Ages was associated with the Rabbanites, was instituted at some point in the ninth century (its attribution to a Hillel in the mid-fourth century is a medieval tradition that has long been disproved in modern scholarship).1  It is a fixed calendar, based on a calculation. Its origins are yet to be fully understood, although some of its elements can be traced back to Talmudic sources. The fragment I present here reveals new ‘missing links’ in the historical development of the calendar calculation, from around the eighth and early ninth centuries.


Antonin A

RNL Ms EVR. Antonin B 154, recto left – from the Collections of the National Library of Russia


Antonin B154 (St Petersburg, Russian National Library) is by no means the earliest fragment of the Cairo Genizah, but it is clearly very early, from the first half of the tenth century or maybe even earlier. It is the outer bifolio of what was a short, probably single-quired compendium of astrology and calendar texts, on parchment, mostly written in Hebrew. Gideon Bohak has found additional folios of this manuscript: New York, JTSA ENA 2689.11-12, and Cambridge, T-S NS 322.64+68.

The compendium begins on recto left of our bifolio, with an astrological text of prognostication based on the position of the moon when the sun enters a Zodiac sign, starting from Libra (which is identified here with the New Year, ‘when God judges the world’). This text occupies a full page. The scribe goes as far as the next Zodiac sign, Scorpio, which he finishes off by writing one last line sideways in the right margin; he stops at Scorpio, for no clear reason.


Antonin B

RNL Ms EVR. Antonin B 154, verso right and left  – from the Collections of the National Library of Russia


The next page (verso right) has a heading: סימן למועדים (‘a sign for the appointed times’), and is entirely about the calendar. At the end, the last three lines begin a new section, now again astrological, listing the seven planets. This introduces a horology of the planetary week, which continues in ENA 2689.11 and then in T-S NS 322.64+68.2

The second half of our bifolio (verso left) continues ENA 2689.12; this suggests that the two ENA folios constituted one bifolio that belonged inside the quire next to Antonin B154. ENA 2689.12 contains an explanation of the calendar calculation in Judeo-Arabic, followed (half-way down the verso) by calendar mnemonic rhymes in Hebrew. The Hebrew text that then straddles ENA 2689.12 (verso) and Antonin B154 (verso left), in a different ink and possibly a different hand, is calendrical and discusses the calculation of the tequfah (equinox or solstice). After that, in yet another hand and in Judeo-Arabic, is an explanation of how to calculate the tequfah by inference from the lunar calendar and the molad (new moon) calculation.

These last two texts on the tequfah, in different hands, are likely to have been added later in a space that was originally left blank. But this is the end of the compendium. The back page of what was the quire (Antonin B154 recto right) is blank.

In the blank space below the end of the compendium (Antonin B154 verso left), a line is drawn and yet another hand records the date and time of birth of a girl called Rachel, probably for horoscopic purposes, with seemingly the birth dates of other children (the ink is very faded), and prayers that God keep them alive and well. This is not a part of the compendium, but the private note of a later user. The date of Rachel’s birth is early morning Shabbat, 10 first Adar, 1250 SE – in our calendar, 2 February 939 CE.3  We can have a reasonable idea of when this note was inscribed: most likely when Rachel was still a child, thus not much later than the middle of the tenth century. The main body of the manuscript, in which this note was inserted by a later user, is necessarily earlier, especially if a period of time is allowed for the last two texts (on the tequfah) to have been added in the interim: certainly the first half of the tenth century, and possibly even earlier.

My attention is drawn to the calendar section in Antonin B154 verso right, which contains two much earlier texts (eighth and early ninth centuries) of great historical importance.


1.    Another primitive rabbinic calendar

Two versions of what I have called ‘primitive rabbinic calendars’, in Aramaic and in Hebrew, have been published to date;4  our fragment contains one of the primitive calendars in its Hebrew version. Primitive rabbinic calendars represent early (and failed) attempts to create a fixed, calculated calendar. They pre-date the institution of the fixed rabbinic calendar in the ninth century, and are probably not much earlier than the eighth.

The main characteristic of primitive rabbinic calendars is that unlike the later, fixed rabbinic calendar, they are not based on a calculation of the new moon (molad), and do not refer to the molad at all. In this respect alone, they lack the sophistication and astronomical accuracy of the later rabbinic calendar. They are based instead on simple calendar rules that can be partially found in the Palestinian Talmud (e.g. the alternation of 29- and 30-day months; the prohibition of the New Year falling on Wednesday or Friday; etc.). The disadvantage of these calendars is that they do not succeed in tracking the moon. In this particular calendar the average year is too short, by one day in less than four years, which rapidly builds up a significant discrepancy and makes the calendar, in lunar terms, completely useless.

The text begins as follows (verso right, line 2):

ואם בא ראש שנה באחד בשבת, בין יש בה עיבור בין אין בה עיבור, חדשים כסדרן

If the New Year falls on Sunday, whether or not the year is intercalated, the months are orderly.5

In the later, fixed rabbinic calendar, the New Year cannot fall on Sunday. This immediately marks out our calendar as ‘primitive’ – the prohibition of Sunday had not yet been introduced.

The phrase ראש שנה without an article is anomalous in Hebrew, but also attested in the primitive calendar of JTSA ENA 1745 (fol. 2 recto, line 8); I have previously argued that this may reflect a translation from the Aramaic ריש שתה.6   Indeed, an Aramaic version of the calendar is preserved in T-S K2.27 and may represent the original language of the primitive rabbinic calendars, even though this manuscript is much later, from the thirteenth century.7 

The text continues along these lines through the other days of the week. For Monday and Saturday, it provides the correct version as in T-S K 2.27.8  But for Thursday in an intercalated year, it says ‘orderly’, which is an error for ‘deficient’.

Meanwhile, additional, later copies of this text have been coming to light: T-S AS 144.119 (recto, perhaps omitting Sunday), and T-S AS 144.292 (recto, omitting Saturday).9  Even though this primitive rabbinic calendar was obsolete and no longer in use by the tenth century, when Antonin B154 was produced, it was still copied and transmitted even much later, as late as the thirteenth century. This demonstrates that literature on the calendar was intended not only for practical purposes, but also as a speculative scientific pursuit – much like the medieval science of Computus among the Christians.

2.    A new ‘missing link’

The second text is later, and a new discovery: it is (by far) the earliest known calendar calculation based on the fixed rabbinic calendar, that is precisely datable, through its worked example, to 842/3 CE.

The fixed rabbinic calendar is based on the calculation of the molad (new moon), using precise astronomical values that were drawn from Ptolemy’s Almagest. These values are expressed in days, hours, and ḥalaqim (1080 parts of the hour). An astronomical lunar month (or ‘lunation’) is 29 days, 12 hours, and 793 parts. The division of the hour in 1080 parts is first mentioned, without explanation, in a piyyut of Rabbi Pinḥas (late eighth century)10  The length of the lunation is then first given by the Muslim scholar Muḥammad ibn Mūsā al-Khwārizmī in his treatise on the Jewish calendar (823/4 CE), but he does not calculate the molad or express any interest in its calculation (it is not even clear that he is describing the fixed rabbinic calendar – but this is for a discussion elsewhere).11  Until now, the earliest attested calendar calculations based on the fixed rabbinic calendar were from the Jewish calendar controversy of 921/2 CE.12  Antonin B154 provides evidence from almost a whole century earlier.

This text is more precisely a calculation of the tequfah (equinox or solstice), based on calculating the difference between solar and lunar years (a lunar year being defined as twelve lunations).


Antonin B 154 verso right, ll. 14-2413 

            הרוצה לידע בכמה בחדש תהיה התקופה מחשב

שנים משנברא העולם ועד עכשיו וטול ימים הנותרים

בין שני חמה לשני לבנה וצרוף אתם ועשה מהם שנים ומה

שאין בהן שנה עשה אתם14 חדשים ומה שאין בו חודש

ראה בכמה ימים וכך התקופה נופלת . לקׄ שנה לחמה טול כׄהׄ

יום דׄ שׄ תׄצׄבׄ חלקים . ולאלף שנים טול רׄנׄאׄ15 יום כׄ שׄעׄ16 שעות

שׄשׄ הׄ וׄ מׄ חׄ17 . לארבעת אלפים שנה יש להן אלף יום זׄ ימיםׄ הׄ18

שעות רׄםׄ חלקים . ולשש מאות יש להן קׄנׄאׄ יום בׄ שעות

תׄשׄצׄבׄ חלקים . וגומלתיה19 אלף קׄנׄחׄ יום יׄבׄ שעות אלף לׄבׄ .

גומלת דׄ אלפים שש מאות גׄ שנים [אלף לׄבׄ גומלת דׄ אלפים

שש מאות גׄ שנים]20 אלף וקׄצׄאׄ יום בׄ21 שעות תׄקׄסׄדׄ חלקים :


Whoever wants to know the date of the tequfah should calculate

the number of years from the creation of the world until now. Take the number of days by which

solar years exceed lunar years and add them up, and turn them into years, and what

does not make up a year turn into months, and what does not make up a month

retain as days, and this is when the tequfah will fall. For 100 years, take for the sun22  25

days, 4 hours, 492 parts. For 1000 years, take 251 days, 20 hours,

300 + 300 i.e. 600 parts. For 4000 years there are 1007 days, 10

hours, 240 parts. And for 600 (years) there are 151 days, 2 hours,

792 parts. Their total is 1158 days, 12 hours, 1032 parts.

The total for 4603 years is [ ….]

1191 days, 4 hours, 564 parts.


Not everything is explained in the text, and I shall only explain what is needed. A solar year (of 365 ¼ days) exceeds twelve lunar months by 10 days, 21 hours, and 204 parts. If for example at the time of the Creation, the spring equinox (a solar event) coincided with the new moon and fell at the beginning of 1 Nisan (Year 1 – as is frequently assumed in earlier rabbinic sources), then one year later, the equinox will occur 10 days, 21 hours, and 204 parts later, thus towards the end of 11 Nisan (Year 2).23  The date of the equinox in any subsequent year can be obtained by adding this value each year. Each time that an intercalation has been made (with the insertion of a second month of Adar), a whole lunation is subtracted from the total. On this basis, the end-result (that gets added to 1 Nisan) should never be higher than 20–30 days.

To save the reader from calculating each year individually, the author simplifies the calculation by providing cumulative values for hundreds and for thousands of years. An illustration of this calculation is given, in the last five lines, with year 4603 from the Creation as paradigm. This enables us to date this composition to 842/3 CE.

The result of the calculation, however, makes no sense at all. 1191 days cannot possibly be right, as it is way above the expected maximum of 20-30 days. It implies that the Jewish calendar has fallen behind the equinox and the seasons by about three years and four months (hence a total of 1191 days), and that Passover is currently falling in the middle of the winter – which is palpably wrong. The author does not explain what to do with this extravagant result, and probably does not know himself. This calculation does not tell us, in the end, the date of the equinox in 843 CE.

The main cause of this gross error is that the calculation is based on a cumulative value that is terribly approximate. it assumes that in 100 years, 36 intercalations are made;24  but this is not always true, and on average the number of intercalations should be a fraction higher (about 36.8), which becomes very significant over 4600 years. In addition, when calculating the last three years (of the year 4603) the author forgets to subtract the intercalation that would be expected in a three-year period. As a result, he misses about 40 intercalary months, which amount to an excessive gap of 1191 days between the equinox and the lunar month.

In later calendar works, the same calculation is executed without any error or approximation on the basis of the 19-year cycle, in which there are exactly seven intercalations, and at the end of which the excess of the sun over the moon is 1 hours, 485 parts, a relatively small number to add up.25  This calculation is simple and always gives the right result. Our author is clearly not aware of this method.

Furthermore, the author does not refer to the molad (new moon) or explain how to calculate it, although this information is essential in this procedure for calculating the lunar date and the precise time of the tequfah.26  Although the author knows the length of the lunation, he may be uncertain about the epoch (starting point) of the molad calculation. Twenty years earlier, in 823/4 CE, al-Khwārizmī was also silent about it.

The author knows the lengths of the solar year and lunar month, but does not have a complete knowledge of the (later) rabbinic calendar calculation. He is experimenting with the values he knows, and not getting the right results. The calculation is sophisticated enough to show that this is not the aberration of a schoolchild or ignoramus: it was the state of the art at the time when this text was composed.

This text reflects an early stage of development of the rabbinic calendar calculation, when the calculated calendar as we now know it was possibly not yet finalized, and when it was certainly not yet in use.27   Antonin B154 thus contributes a rare ‘missing link’ to the history of the rabbinic calendar in its formative period.



1 H. Y. Bornstein, ‘Divrei yemei ha-ibbur ha-aḥaronim’, Ha-Tequfah, 14–15 (1922), 321-372, and 16 (1922), 228-92; S. Stern, Calendar and Community: A History of the Jewish Calendar 2nd century BCE – 10th century CE, Oxford: Oxford University Press, 2001, 175-210.

2 These astrological texts are due to be edited by Gideon Bohak.

3 This is one of the earliest attested uses of the Seleucid era in the Cairo Genizah. See E. Krakowski and S. Stern, ‘The ‘oldest dated document of the Cairo Genizah’ (Halper 331): The Seleucid era and sectarian Jewish calendars’, Journal of the Royal Asiatic Society, 31 (2021), 617-634, on p. 628, although we were unaware of the date in this fragment.

4 S. Stern, ‘A primitive rabbinic calendar text from the Cairo Genizah’, Journal of Jewish Studies, 67 (2016), 68-90, on T-S K 2.27, discovered by Nadia Vidro; and S. Stern, ‘New light on the primitive rabbinic calendars: JTS ENA 1745’, Journal of Jewish Studies, 69 (2018), 262-279, on a manuscript that I subsequently discovered. Gideon Bohak first drew my attention, in 2018, to the primitive rabbinic calendar in Antonin B154. I am as grateful as ever to my generous colleagues.

5 The ‘months’ are Marḥeshwan and Kislew, which are of variable length; ‘orderly’ means they count 29 and 30 days (respectively), in contrast to ‘deficient’ (both 29 days) and ‘full’ (both 30 days). This information, together with the day of the week on which the New Year falls, is sufficient to construct the calendar for the whole year.

6 Stern (2018) 264.

7 As datable palaeographically: Stern (2016) 71.

8 The version of ENA 1745 appears erroneous in both these days: see Stern (2018) 267 n. 27, 274-5.

9 Discovered by Yosef Yizḥaq Rottenberg in October 2021; my thanks for sharing these references. 

10 Ms HUC Acc_975, edited by S. Elizur, Piyyutei Rabbi Pinḥas ha-Kohen, Jerusalem: WUJS, 2004, 643 (no.104, ll. 24-5).

11 Ms Bankipore 2468, fols. 115r-117r (Khuda Bakhsh Library, Patna-Bankipore, India). There is no good edition of this text, but a Hebrew translation is in T. Langermann, 'Ematay nosad ha-luaḥ ha-ivri?’, Asufot 1 (1987) 159-168.

12 S. Stern, The Jewish Calendar Controversy of 921/2 CE, Leiden: Brill, 2019.

13 Punctuation and spacing are mine, except for the colon at the end.

14 The aleph is pointed, perhaps for disambiguation.

15 The text reads דׄנׄאׄ, a scribal error (similarly below, דׄםׄ for רׄמׄ, and at the end, תקסר for תׄקׄסׄדׄ).

16 For שעות, a dittography.

17 For הוא וׄ מאות חלקים. The text appears to read הׄוׄמׄוׄ, but I consider the last character to be an error for ḥet, or even a very distorted ḥet (note the dogleg in the upright stroke). The mem has a dagesh, I am not sure why. More generally I do not understand why this phrase is so heavily abbreviated, and why the author did not write instead of all this, as elsewhere, חלקים. Perhaps something went wrong in the process of copying.

18 Error for יׄ (10)

19   GML is a loan from the Arabic. Nadia Vidro suggests that this is a phonetic spelling of Ar. jumlatihi, ‘its sum’.

20 In brackets: dittography.

21 Error for דׄ (4)

22 i.e. ‘for the excess of the sun over the lunar years’. לקׄ שנה לחמה טול cannot be translated as ‘for 100 solar years, take …’, because the years from the Creation that have been counted are presumably in the Jewish calendar, which is lunar (or ‘lunisolar’). Moreover, ‘solar years’ are called שני חמה in the third line

23   More precisely, this value needs to be added to the molad of Nisan Year 2, which (in this case) could easily push the date of the equinox into 12 Nisan. The author does not explain this, and possibly does not fully understand it either.

24 The value given here for 100 years is 25d 4h 492p, which is obtainable from subtracting the total of 36 lunations from 100 times the solar excess of 10 days, 21 hours, and 204 parts.

25 This method is used, for example, in the Judeo-Arabic text here on verso left (l. 10), which is presumably later.

26 See above, note 23.

27 As demonstrated from the letter of an exilarch dated 835/6 CE: T-S 8G7.1 (edition and discussion in Stern 2001: 180-1, 184-6, and 277-83).

I am grateful to Nadia Vidro and Gideon Bohak for their comments on much earlier drafts.

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