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The time units of traditional Hebrew calendar arithmetic are years, months, days, hours, and parts (chalakim). One part (chelek) equals 10/3 seconds = 3+1/3 seconds = 1/18 of a minute = 1/1080 of an hour = 1/25920 of a day.
This web page examines where this strange time unit comes from, and highlights some of its noteworthy properties.
Rabbi Moshe ben Maimon ("Rambam"), also commonly known by his Greek name, (Moses) Maimonides, wrote a book entitled Hilchot Kiddush haChodesh (title translated as "Sanctification of the New Month" or alternatively as "Sanctification of the New Moon") around the Julian year 1178 or Hebrew year 4938, which is one of the treatises in his Mishneh Torah collection (code of Jewish Law). In this book, in chapter 6 item 2, Rambam wrote that the number 1080 parts per hour was chosen because it is divisible without remainder by 2, 4, 8 and 3, 6, 9 as well as 5 and 10. This is true, but it is also true for the smaller number 360, which equals 1080/3.
All of the divisors of 1080 are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540, and 1080.
All of the divisors of 360 are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.
Of all of the above divisors, only 1, 2, 3, and 5 are prime numbers.
The number 1080 derived from prime numbers = 23 × 33 × 5.
The number 360 derived from prime numbers (omitting one power of 3 since 360 is 1/3 of 1080) = 23 × 32 × 5.
Actually, the length of a part is fixed by the traditional molad interval.
On page 25a of the Talmud Bavli tractate Rosh HaShanah, Rabban Gamliel is quoted as saying "Thus have I received a tradition from the house of my father's father: the rebirth of the Moon is not less than twenty-nine and half days, two-thirds of an hour, and seventy-three parts from the previous one." That is, 29 days, 12 hours, 40 minutes, and 73 parts. Expressed in terms of days with an exact fractional remainder that equals:
29 + 1/2 + 40/(24 × 60) + 73/(1080 × 24) days
= 29 + 1/2 + 40/1440 + 73/25920 days
= 29 + 13753/25920 days
This exactly matches the figure that was published by Ptolemy in his voluminous astronomy text, Almagest (literally "The Great Book", the name assigned to it by Arab translators centuries later), but Ptolemy cited Hipparchus as the source for the mean duration of the lunation (average interval between successive matching lunar phases). Ptolemy gave the mean lunation length (mean synodic month) in sexagesimal, the customary base 60 numeral system used by the ancient Babylonians (since the Seleucid era after 312 BC):
mean synodic month = 29 days 31' 50'' 8''' 20''''
That is, 29 days, 31 minutes, 50 seconds, 8 thirds, and 20 fourths. Expressed in base 10 numerals in terms of days with an exact fractional remainder that equals:
29 + 31/60 + 50/(60 × 60) + 8/(60 × 60 × 60) + 20/(60 × 60 × 60 × 60) days
= 29 + 31/60 + 50/3600 + 8/216000 + 20/12960000 days
= 29 + 13753/25920 days
The denominator 25920 of the fractional remainder is the number of parts in a day! The number of parts per hour = 25920/24 = 1080, and so we see how a part equals 1/1080 of an hour. Since there are 60 minutes per hour there are 1080/60 = 18 parts per minute or 1/18 of a minute per part. Likewise there are 60 seconds per minute so there are 60/18 = 10/3 = 3+1/3 seconds per part.
The numerator 13753 and denominator 25920 of the fractional remainder imply an underlying mean lunar cycle of mostly alternating 30-day and 29-day months, spread as smoothly as possible in a cycle having a total of 25920 months, in which 13753 months have 30 days and the remaining 25920 – 13753 = 12167 months have 29 days, which would yield a mean month of exactly 29+13753/25920 days. The exact duration of such a cycle is 2095+31/47 mean Hebrew calendar years.
In any lunar cycle, which must contain an integer number of days, 30-day months must occur slightly more frequently than 29-day months, such that 2 consecutive 30-day months occur at intervals of either 17 or 15 months, where the 17-month interval is approximately twice as common as the 15-month interval. The shortest possible such cycles that have a mean month closest to that of the traditional molad interval are a 1144-month cycle having a slightly longer mean month of 29+607/1144 days = 29 12h 44m 3+51/143s, with a "part" duration of 1m 15+75/143s, and a 1095-month cycle having a slightly shorter mean month of 29+581/1095 days = 29 12h 44m 3+21/73s and a "part" duration of 1m 18+66/73s. Thus a cycle that is substantially shorter than the traditional molad cycle could have been selected while maintaining comparable accuracy. Note that the duration of a "part" has nothing to do with the divisors of the number of parts per day, nor with the precision of time resolution, rather it is simply a consequence of the exact fractional expression of the mean month, where the numerator is the number of full months per cycle and the denominator is the total number of months per cycle, hence the denominator determines the number of "parts" per day.
Due to on-going tidal transfer of angular momentum from Earth to Moon, the mean lunar cycle gets progressively shorter (in terms of mean solar days), so that for the present era the shortest lunar cycles having accurate mean months are an 850-month cycle with a slightly-too-long mean month of 29+451/850 days = 29 12h 44m 2+14/17s and a "part" duration of 1m 41+11/17s, and an 801-month cycle with a slightly-too-short mean month of 29+425/801 days = 29 12h 44m 2+62/89s and a "part" duration of 1m 47+77/89s. If one must choose a fixed lunar cycle then to minimize future molad drift and maximize the calendrically useful longevity it is always better to intentionally choose a cycle that has a slightly-too-short mean month.
The traditional molad interval exactly matches the length of the mean lunar cycle given by Rebbe Eliezer ben Hurcanus, traditional author of Pirkei D'Rebbi Eliezer, who described the molad cycle in considerable detail (chapters 6 and 7) and lived in approximately the same era as Ptolemy, although his teachings weren't assembled and published until several centuries later.
The duration of a part or chelek equals the earlier Babylonian barleycorn (pronounced she), the smallest Babylonian time unit, which was 1/72 of a time degree. The time degree was the principal Babylonian unit of time, corresponding to the time required for one degree of motion of Sun across the meridian = 1/360 of a solar day = 1440 minutes per day/360 time degrees per day = 4 minutes per time degree. Thus 4 minutes divided by 72 = 1/18 of a minute = 1 chelek. The time degree also very nearly equals the difference in duration between the solar day and sidereal day, which in the present era amounts to about 3 minutes and 55.9 seconds. The Babylonian finger was 6 barleycorns = 1/12 of a time degree = 1/3 of a minute = 20 seconds of time. The cubit was 180 barleycorns = 5/2 time degrees = 10 minutes of time. The hour itself, corresponding to 15 time degrees, was a Seleucid time unit that was probably obtained from Egypt. The Babylonian beru or double hour, corresponded to 30 time degrees. The mean synodic month in Babylonian time units was 29 days, 6 double hours, 11 time degrees, and 1 barleycorn.
If we divide the numerator 13753 by 1080 to separate the number of hours from the remaining parts we obtain:
The remaining 793 parts is the same as 2/3 hour + 73 parts = 44 minutes + 1 part = 44 + 1/18 minutes. It is exactly one part greater than 2/3 + 1/15 = 11/15 of an hour.
In one half of the complete molad cycle there are 25920/2 = 12960 months, so each complete cycle contains an excess of 13753 – 12960 = 793 full months, which corresponds exactly to the remaining 793 parts in excess of 29 + 1/2 days. Likewise there is a deficiency of 12960 – 12167 = 793 deficient months, since that number of otherwise deficient months are made full in each complete cycle.
Therefore the traditional molad interval is 29 days, 12 hours, 793 parts. The duration of the molad interval is critical to the traditional Hebrew calendar arithmetic, and must be expressed using only whole numbers and proper fractions so that any date can be calculated exactly and unambiguously. (Long after posting this web page, I learned that the Vilna Gaon similarly explained that only the division of hours into 1080 parts allows the duration of the lunation to be expressed without use of a fraction, see the Kol Eliyahu commentary on Talmud Bavli tractate Rosh HaShanah page 25a.) Exact calculation of a molad moment isn't as complicated as it might seem, because, as pointed out by Rambam (chapter 6) the 29 days comprise 4 weeks plus 1 day remainder, therefore if one already knows the molad moment for a given month then the molad moment of the next month will be 4 weeks, 1 day, 12 hours, and 793 parts later. Similarly, if one knows the molad moment for Tishrei of a given year (used in determining the date of Rosh HaShanah), then the next molad of Tishrei after a non-leap year will be 12 × (4 weeks, 1 day, 12 hours, and 793 parts) = 354 days and 9516 parts = 50 weeks, 4 days, 8 hours, and 876 parts (73/90 of an hour = 48+2/3 minutes = 48 minutes and 40 seconds) later, and after a leap year will be 13 × (4 weeks, 1 day, 12 hours, and 793 parts) = 383 days and 23269 parts = 54 weeks, 5 days, 21 hours and 589 parts (32 minutes and 13 parts) later.
By contrast, in the decimal representation of the exact value of the traditional molad interval = 29.530594135802469... days, the overscored digits repeat forever. To carry out exact molad moment calculations using floating point arithmetic, for example using a computer or a calculator, after obtaining the floating point result set aside the integer part to the left of the decimal point (the day number), multiply the fractional value by the number of parts in a day (25920), and then round that value to an integer to obtain the number of parts to that molad moment within the set-aside day number.
The number 793 is divisible by 1, 13, 61, and 793. As the traditional molad interval is added for each elapsed month the remaining number of parts in the molad moment (after taking away the days, hours, and minutes) increments by one for each elapsed month. This means that the traditional molad completes a full cycle of all possible molad moment times (ignoring the weekday) in as many months as there are parts in a day = 25920 months.
There are 235 months per 19-year cycle of the Hebrew calendar. The divisors of 235 are 1, 5, 47, and 235, whereas 19 is a prime number. The average number of months per traditional Hebrew calendar year = 235/19, so the number of years in the molad moment repeat cycle = 25920 / (235/19) = 2095+31/47 years. Every 18 months the molad moment is exact to the minute with zero parts remaining. Every 1080 months or 87+7/22 years the molad moment is exact to the hour with zero minutes and zero parts remaining, which happened most recently for the traditional molad moment of Cheshvan in year 5765. A full cycle of every possible molad moment landing on every possible weekday takes 7 × 25920 = 181440 months / (235/19) = 689472/47 = 14669+31/47 years (this is to be expected because 181440 is the number of parts per week). A full cycle for the molad of Tishrei of the first year of the 19-year cycle to go through every possible molad moment landing on every possible weekday takes 47 times longer, or 689472 years.
The total number of parts per molad interval = 1080 × ( 29 × 24 + 12 ) + 793 = 765433 parts, which can be written as the descending sequence 765432+1.
According to Rambam's book as cited above (chapter 10, #1) there are 76 regaim (moments) in a chelek (part). Why is there such a seemingly strange time unit in our tradition? Once again, the purpose was to enable calendar calculations to be carried out with integers.
The molad interval fixes the Hebrew calendar mean year length at 235 lunar months times the duration of the molad interval divided by 19 years = 365 days 5 hours 55 minutes and 7+12/19 chalakim. Use of the rega time unit eliminates the fractional 19ths of a chelek, which become 76 × 12/19 = 48 regaim, but if that was the only reason then why did the sages make the rega 4 times smaller than 1/19th of a chelek?
It was done so that when the solar year was divided into 4 equal seasons there would not be any fractional remainder = 91 days 7 hours 28 minutes 15 chalakim and 31 regaim per season. For more information about this traditional Jewish method for approximation of equinox and solstice moments, see the heading "Method #2: Tekufat Adda" on my "Rambam and the Seasons" web page.
Given that a chelek is 10/3 seconds, a rega is only 10/3 / 76 = 5/114 of a second = only 43+49/57 milliseconds!
Knowing that, the next time somebody tells you to "wait a moment" (in Hebrew chakae rega), it will be proper if you immediately reply "done".
Given that a chelek is 1/1080 of an hour, a rega is only 1/1080 / 76 = 1/82080 of a hour.
Obviously, although today's electronic instruments have no problem measuring time intervals as short as picoseconds (trillionths of a second), in the era when the rega time unit was conceived there was no possibility of actually measuring a duration as short as a rega, so its existence was entirely theoretical, although some sages of the Talmud described it as the blink of an eye. (Both the Talmud Yerushalmi and the Talmud Bavli give several incompatible opinions on the exact duration of a rega, which I will not discuss here.)
The rega also has no practical significance for its intended purpose in calculation of the length of the solar year or of the solar seasons, because the year-to-year variations in solar year length amount to ±15 minutes, mainly due to the varying influence of Moon on Earth's position, and because the actual astronomical season lengths are not equal and in fact routinely differ by several days. For further information see "The Lengths of the Seasons" at <http://www.sym454.org/seasons/>.
In terms of Atomic Time, the Mean Synodic Month (MSM) is getting steadily longer because tidal forces are accelerating Moon further away from Earth at the mean rate of about 38 mm per year. However, Earth's rotation rate is slowing down more significantly, also due to tidal forces, so that in terms of mean solar days, which are all-important for calendar calculations, the net effect is that the MSM is actually getting shorter. In other words, tidal forces result in the transfer of angular momentum from Earth to Moon. The net result is that the mean interval from New Moon to New Moon gets steadily shorter relative to the mean solar day, presently at the rate of about 27+1/3 microseconds or 41/5 "microparts" per lunar month.
For a detailed analysis of the gradual long-term changes in the length of the mean lunar cycle, please see my web page "The Length of the Lunar Cycle" at <http://www.sym454.org/lunar/>.
In the era of Hillel ben Yehudah, who is said to have established the traditional fixed arithmetic Hebrew calendar around Hebrew year 4119, the MSM was only a few milliseconds shorter than the traditional molad interval, but today the MSM is almost 3/5 of a second shorter, or 9/50 of a part.
At the time of the Exodus from Egypt, traditionally given as Hebrew year 2448, the mean synodic month was about 2/5 second longer than the traditional molad interval.
It seems certain that the Babylonians were the astronomers who originally determined the traditional length of the mean lunation interval, but that interval was about 1/4 second too short for their own era, possibly due to rounding its duration to the nearest barleycorn.
Although, as mentioned above, Ptolemy cited Hipparchus as the source for his sexigesimal value for the lunation length, we have no existing hardcopy of any such work by Hipparchus. That interval was accurate for the era of Hipparchus, which was the same era as the Maccabees in Israel.
For more information about the difference between fixed length molad interval and the actual progressively shorter length of the lunar cycle please see my web page "Moon and the Molad of the Hebrew Calendar" at <http://www.sym454.org/hebrew/molad.htm>, in particular the topic heading "Which Meridian Does the Molad Moment Refer To?" and the chart that is there.
The Hindu synodic month used for the old Hindu calendar before 1100 AD, as documented around 499 AD in the Arya Siddhanta, was 29+2362563/4452778 days = 29 days 12h 44m 2+597062/2226389s (the fraction is just over 1/4 second), or just over a second shorter than the astronomical mean synodic month when Hindus started to use that calendar. By contrast, the mean synodic month used for the modern Hindu calendar, as documented in the Surya Siddhanta around 1000 AD, is 29+7087771/13358334 days = 29 days 12h 44m 2+1777862/2226389s (the fraction is almost 4/5 second), which was almost 1/3 second too short for the era when Hindus started to use that calendar, but is actually essentially perfect for the present era.
Anyone who is designing a fixed cycle lunar calendar for use today and for as long as possible into the future would be well advised to choose a mean lunation period that is intentionally slightly too short, to maximize the duration that the calendar will serve with reasonable accuracy, because the astronomical mean synodic month is getting progressively shorter in terms of the mean solar days that calendars need.
The ancient Babylonians used sexagesimal numerals because they had a special preference for the numbers 6, 10, 60, 360, and so on. They were the originators of measuring angles in degrees (360° in a circle, or Sun appearing to move 360° in a day), arcminutes (60' per degree), arcseconds (60'' per arcminute) and so on.
Interestingly, if one takes away just a single part from the traditional molad interval, the number of parts per molad interval has the descending sequence 765432 parts, and the fractional part of a day in excess of 29 days reduces as follows:
29 + 13753/25920 – 1/25920 = 29 + 13752/25920 = 29 + 191/360 days
How much happier would the Babylonians have been to see the number 360 in the denominator! Perhaps that was actually their original estimate of the lunation interval, no doubt based on solar and lunar eclipse studies, but later it must have been found necessary to "kick it up a notch" for improved accuracy.
Around Hebrew year 13980, more than 8 millennia after the present era, the shorter interval of 29 + 191/360 days will very nicely suit the actual mean duration of the lunation. For further details explaining this estimate of the future shorter mean lunation interval, please see the progressive molad of the Rectified Hebrew calendar at <http://www.sym454.org/hebrew/rect.htm>.
The slaughter and sacrifice of the Paschal lamb (Korban Pesach) was to take place in the afternoon of the 14th of Nisan, starting 1/2 hour after noon, that is nominally 18+1/2 hours after sunset = 14th of Nisan plus 37/48 of a day. The number 14+37/48 divided by 29 days, 12 hours, 793 parts is almost exactly 1/2, that is 1/2 of a traditional molad interval. The 14th of Nisan is actually the 13th day of the month, because the day of month numbering starts from day one not day zero, but the first visible new lunar crescent could be seen no earlier than one day after the actual lunar conjunction. Therefore, although relative to sunset at the start of Nisan the Korban Pesach moment was one day earlier than 1/2 of a molad interval, that moment ought to have been very close to the actual full moon moment (which of course can occur at any time of day). At the next sunset, nominally 5+1/2 hours after the Korban Pesach, the Paschal Moon near the eastern horizon will appear full to visual observation.
Atmospheric refraction makes objects (most notably Sun and Moon) near the horizons appear about 1/2° higher in the sky than their true geocentric positions. Atmospheric refraction is greater when the air temperature is colder or pressure is higher. Topocentric lunar parallax (viewing Moon from the surface of Earth) makes Moon appear about 1° lower than its true geocentric position. Taken together, these effects essentially offset each other, so that at non-polar latitudes we can say that Moon is Full if it is rising when Sun is setting, or if Moon is setting when Sun is rising. It would not often happen that these events coincide so closely in time. To allow for variable timing, we can say that at non-polar latitudes if the Moon rise/set is within about 30 minutes of the Sun set/rise, respectively, then there will not be another Moon rise/set that is closer to Full Moon during that lunar month.
The actual Full Moon moment is observationally best defined as the maximum of a lunar eclipse, regardless of where in the sky the observer sees it. The moment of the true astronomic maximum is observationally difficult to determine without instrumentation, however, because that is the moment of minimum distance between the center of Earth's umbral shadow and the center of Moon. There are historical reports of cases where a lunar eclipse was in progress at a moonrise just prior to the beginning of sunset (both Sun and Moon appeared just above their respective horizons). This uncommon arrangement is possible because at the distance of Moon the umbra of Earth's shadow has a diameter that is more than 3 times wider than the lunar diameter — greater when Moon is near perigee or Earth is near perihelion. The duration of a lunar eclipse is maximal when the Full Moon is near either of the lunar orbital nodes — in such cases Moon crosses the widest diameter of Earth's shadow.
Globally, seeing a lunar eclipse in progress at sunset / moonrise or sunrise / moonset should not be highly exceptional, in fact every lunar eclipse ought to appear that way to all locales along the sunset terminator or sunrise terminator, respectively.
Having just discussed the observational criteria for Full Moon, I can't resist the following diversion:
The first tractate of both the Babylonian and Jersualem Talmud is Berachot (Blessings), and both start out by discussing the question of when does one calendar day end and another begin, and what is the duration of uncertainty (twilight) between days? The Hebrew term used for twilight is bein hashemashot, which literally means "between the Suns". This issue is of concern with regard to when is the earliest moment that certain night-time blessings be said, when does a day of fasting or high holy day or sabbath end, when does a cohen (priest) become fit for duty, and on which day should a circumcision be performed when the boy was born around the time of sunset?
In the Jerusalem Talmud, on page 3 or 4 (depending on the printed edition), Rabbi Yehuda ha-Nasi is quoted as having said that when Moon is just about to rise and Sun is just about to set, that is bein hashemashot. From the commentaries and the subsequently quoted comments, it seems that his statement has been widely misunderstood. The above discussion concerning observational criteria for the Paschal Moon will tie into the following explanation.
Rabbi Chaninah is next quoted as having said that Rabbi Yehuda ha-Nasi meant to say that bein hashemashot is when Moon is just about to rise and Sun has just fully disappeared from view. Some commentators say that Rabbi Chaninah was arguing with or correcting Rabbi Yehuda ha-Nasi, but others say that that is impossible because Rabbi Chaninah was a later authority, and in Jewish Law a later authority is almost always assumed to be the lesser authority. Others say that Rabbi Chaninah was not contradicting his superior, but rather attempting to emend or clarify the comments of his teacher.
Amorah Shmuel "the astronomer" next is quoted as agreeing with Rabbi Chaninah, and he added that it is impossible at Full Moon to see both Moon and Sun above the horizon at the same time.
Bearing in mind what we learned about the observational criteria for the Paschal Moon, above, we can say that Rabbi Yehuda ha-Nasi described the scenario of a Full Moon occurring at the same time as sunset at the observer's locale. All of the following situations are observationally equivalent in that regard:
In all of the above cases, all of the land that the observer can see is in-between (bein) Sun and Moon, which are at opposite extremes in position (at opposition), and both Sun and Moon appear to be full discs. Moon is thus Sun-like in shape and also near its maximum brightness. At such moments it is as if the world is "between two Suns", hence the derivation of the term bein hashemashot! Thus we see that Rabbi Yehuda ha-Nasi gave a valid definition explaining the origin of that term, and there was no need for Rabbi Chaninah or Amorah Shmuel to add anything to what Rabbi Yehuda ha-Nasi said.
The scenario described by Rabbi Chaninah corresponds to a moment slightly after Full Moon, when Moon has moved beyond opposition. In that lunar month, however, there will not be another sunrise or sunset at which Moon will appear closer to opposition. Both of the following are observationally equivalent to his scenario:
The maximum of a lunar eclipse is a definitive indication of the actual Full Moon moment when Moon reaches opposition. The scenario described by Amorah Shmuel, which he said was not possible at Full Moon, actually can be observed, in fact there have been well-documented instances where a lunar eclipse was observed with both Sun and Moon above their opposite horizons. In fact, at essentially every lunar eclipse there ought to be a meridian of longitude where the eclipse is in progress just after sunrise and just before moonset, and simultaneously another meridian of longitude where the eclipse is in progress just before sunset and just after moonrise. This is possible because of atmospheric refraction of light, which makes objects near any horizon appear higher than their actual geometric positions. Alternatively, such an observation corresponds to a moment slightly before Full Moon, when Moon has not yet moved into opposition. Again, in that lunar month there will not be another sunrise or sunset at which Moon will appear closer to opposition. Both of the following are observationally equivalent to his scenario, in which both Moon and Sun are just above their respective horizons:
It would seem that Rabbi Chaninah and Amorah Shmuel in effect expanded the boundaries around the original statement of Rabbi Yehuda ha-Nasi, akin to but much more restrictive than my half-hour "leeway" suggested in the discussion of the Paschal Moon, above, but did not enhance the explanation of the origin of the term bein hashemashot. This little debate in the Talmud, however, in no way defines or modifies the highly controversal criteria for the start and end moments of twilight itself, for the purposes of Jewish Law.
Updated 10 Sivan 5773 (Traditional) = 10 Sivan 5773 (Rectified) = May 21, 2013 (Symmetry454) = May 19, 2013 (Symmetry010) = May 19, 2013 (Gregorian)