The following is a compilation of interesting facts or information about the moon that I found on my walk to chodesh. Most of this I did not know and I suspect the average person on the street most likely would not know either. As such most people should find the following moon information interesting. A lot of it is approximate. Except for the two months quoted, a lot of values were determined using models that are approximate. Probably nothing should be quoted from here as exact. Anyone wanting more exact information should probably go to an actual astronomy site. For everyone else, enjoy!

Moon phases
Moon phases

There are mainly 8 phases:
New Moon, Waxing Crescent, First Quarter, Waxing Gibbous, Full Moon, Waning Gibbous, Last Quarter, and Waning Crescent. Shown here with some repeated to give an idea of New Moon to New Moon. Also the view shown here is for the southern hemisphere.

The synodic month of 29.53 days
The synodic month is the length of time it takes for the moon to travel from one phase to the same phase in the following month. Commonly defined as from new moon to new moon. It is approximately 29.53 days.

A synodic month is 29.53059 days (29 days, 12 hours, 44 minutes, 2.8 seconds) and is measured from New Moon to New Moon.
http://en.wikipedia.org/wiki/Lunar_month

The sidereal month of 27.32 days
The sidereal month is the length of time it takes the moon to complete one orbit around the earth. It is approximately 27.32 days. That is, it takes the moon approximately 27.32 days to orbit the earth.

The Moon's orbital period in a non-rotating frame of reference (which on average is equal to its rotation period) is about 27.32166 days (27 days, 7 hours, 43 minutes, 11.6 seconds). This is known as a sidereal month and is measured by observing how long it takes the Moon to pass a fixed star on the celestial sphere.
http://en.wikipedia.org/wiki/Lunar_month


Why are these two months different? If it takes the moon 27.32 days to orbit the earth, then why does it take approximately 29.53 days to go from new moon to the next new moon?
Note the following diagram [nothing to scale]:

Moon orbit

The earth moves 29.1 degrees around the sun in one synodic month
For the time it takes the moon to travel from one new moon to another, that is 29.53 days, the earth moves approximately 29.1 degrees in its orbit around the sun. The new moon occurs when the moon is roughly between the earth and the sun. For this to happen, using simple rules of mathematics, the moon must also travel an extra 29.1 degrees in its orbit around the earth. This is why the synodic month is longer than the sidereal month. The extra travel takes about 2.2 days so the sidereal month can be found by subtracting 2.2 from 29.53 giving approximately 27.33 days [27.32 is more correct].
Note: the sidereal moon position shown above would be of course at approx 27.32 days.

Orbit of earth about sun nearly circular
The above calculations were determined by approximating the orbit of the earth about the sun as circular. The orbit of the earth is elliptic but nearly circular:

The eccentricity of the Earth's orbit is currently about 0.0167, meaning that the Earth's orbit is nearly circular, the semiminor axis is 98.6% of the semimajor axis.
http://en.wikipedia.org/wiki/Orbital_eccentricity

The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0<e<1). When the eccentricity is 0 the foci coincide with the center point and the figure is a circle.
http://en.wikipedia.org/wiki/Ellipse

The angle of travel for the earth about the sun was found by approximating the degrees given the synodic month of 29.53 days as follows:

Calculation 1

An approximate value for the sidereal month can be determined using the synodic month by working out one orbit of 360 degrees given the extra 29.1 degrees about the earth needed for the moon to complete a synodic month as follows:

Calculation 2

Similarly the difference in days between the two months can be determined by finding the time for the moon travelling 29.1 degrees about the earth as:


Calculation 3

Actual orbit of earth about sun not elliptic or circular
The earth and moon orbit the sun about their centre of mass:

the moon does not orbit the exact center of the Earth, but a point on a line between the center of the Earth and the Moon, approximately 1,710 km below the surface of the Earth, where their respective masses balance. This is the point about which the Earth and Moon orbit as they travel around the Sun.
http://en.wikipedia.org/wiki/Center_of_mass#Astronomy

As such, technically the orbit of the earth would be along a slightly wobbly path as the centre-of-mass of the earth-moon system follows the elliptic path of orbit. For the above calculations the earth orbit path was approximated as circular.

Earth-moon scale usually not correctly shown
The Moon is approximately 3500 km wide.
The Earth is approximately 12800 km wide.
The center-to-centre distance is approximately 384,000 km.
You usually see nice sketches of the Earth and the Moon and they are usually pretty close. To scale the reality is more like the following:


earth moon scaled picture

Every synodic month not exactly 29.53 days
Not exactly true: the odd month could turn out to be exactly 29.53 days but more often it will be more or less. The value of 29.53 days is an average over quite a few months. For 2011 the range appears to be approximately 10 hours. So through 2011 a more correct value could be quoted as about 29.53 days ± 5 hours. The following data worked for 2011 shows this:

2011 month periods data

2011 month periods plot

Through 2011 the synodic month length ranged from 29.34 days up to about 29.76 days.
The maximum value minus the minimum value is about 0.42 days or approximately 10.1 hours.
So we could say that the period for the synodic month any month for 2011 could be considered to be approximately 29.53 days plus or minus 5 hours. It could be plotted. Interestingly if you could determine approximately where you were on the graph, you could have a good guess at the next moon phase date you were investigating. For anyone doing visual sightings this graph could be extremely important.
Note: the synodic month being shorter than most calendar months will sometimes fall inside a month: there were two new moons in July, hence Jul-Jul on the axis.
Also Jerusalem times were used for the astronomical new moon to ascertain the above periods.

From year to year this range probably won't change very much but consider the following quote:

Because of perturbations in the orbits of the Earth and Moon, the actual time between lunations may range from about 29.18 to about 29.93 days. The long-term average duration is 29.530589 days (29 d 12 h 44 min 2.9 s).
http://en.wikipedia.org/wiki/Month#Synodic_month

This gives a worst possible range of 0.75 days or 18 hours. This may be extreme and for 2011 only about a 10 hour range occurred. Still this may mean that a close watch on the synodic month lengths should be given each year for those who are interested in the lunar months.

Why don't we have a solar eclipse every new moon?
The time of the astronomical new moon occurs when the moon passes between the earth and the sun. If we have a new moon every 29.53 days why don't we have a solar eclipse then too?

The Moon is a cold, rocky body about 2,160 miles (3,476 km) in diameter. It has no light of its own but shines by sunlight reflected from its surface. The Moon orbits Earth about once every 29 and a half days. As it circles our planet, the changing position of the Moon with respect to the Sun causes our natural satellite to cycle through a series of phases:

      • New Moon > New Crescent > First Quarter > Waxing Gibbous > Full Moon > Waning Gibbous > Last Quarter > Old Crescent > New Moon (again)

The phase known as New Moon can not actually be seen because the illuminated side of the Moon is then pointed away from Earth. The rest of the phases are familiar to all of us as the Moon cycles through them month after month. Did you realize that the word month is derived from the Moon's 29.5 day period?

To many early civilizations, the Moon's monthly cycle was an important tool for measuring the passage of time. In fact many calendars are synchronized to the phases of the Moon. The Hebrew, Muslem and Chinese calendars are all lunar calendars. The New Moon phase is uniquely recognized as the beginning of each calendar month just as it is the beginning on the Moon's monthly cycle. When the Moon is New, it rises and sets with the Sun because it lies very close to the Sun in the sky. Although we cannot see the Moon during New Moon phase, it has a very special significance with regard to eclipses.

An eclipse of the Sun (or solar eclipse) can only occur at New Moon when the Moon passes between Earth and Sun. If the Moon's shadow happens to fall upon Earth's surface at that time, we see some portion of the Sun's disk covered or 'eclipsed' by the Moon. Since New Moon occurs every 29 1/2 days, you might think that we should have a solar eclipse about once a month. Unfortunately, this doesn't happen because the Moon's orbit around Earth is tilted 5 degrees to Earth's orbit around the Sun. As a result, the Moon's shadow usually misses Earth as it passes above or below our planet at New Moon. At least twice a year, the geometry lines up just right so that some part of the Moon's shadow falls on Earth's surface and an eclipse of the Sun is seen from that region.
http://www.mreclipse.com/Special/SEprimer.html

Admittedly we only needed the last paragraph but there was a wealth of information here that just could not be missed. Curiously though this site has the orbit of the moon as the synodic length, not the sidereal. A simple mistake.

Moon and month are cognates
Moon and month are words that have a common origin. They are related:

A month is a unit of time, used with calendars, which was first used and invented in Mesopotamia, as a natural period related to the motion of the Moon; month and Moon are cognates. The traditional concept arose with the cycle of moon phases; such months (lunations) are synodic months and last approximately 29.53 days.
http://en.wikipedia.org/wiki/Month

In linguistics, cognates are words that have a common etymological origin. The word derives from the Latin cognatus (blood relative).
http://en.wikipedia.org/wiki/Cognate

The time it takes the Moon to go through all its phases is about a month, and that was so important to our ancestors that they created the period of time we call a month. Maybe you've even noticed that the word month is like the word moon.
http://spaceplace.nasa.gov/en/kids/phonedrmarc/2004_march.shtml

Synodic month generally defined as being from new moon to new moon
Just about everywhere you look the lunar synodic month is defined as being from new moon to new moon; all the way from NASA down to the most general sites on astronomy:

the mean synodic month (New Moon to New Moon)
http://eclipse.gsfc.nasa.gov/phase/phases2001.html

The New Moon phase is uniquely recognized as the beginning of each calendar month
http://www.mreclipse.com/Special/SEprimer.html

A synodic month is 29.53059 days (29 days, 12 hours, 44 minutes, 2.8 seconds) and is measured from New Moon to New Moon.
http://en.wikipedia.org/wiki/Lunar_month

Lunar Month: The period between successive new moons (29.531 days).
TheSage Dictionary/Thesaurus.

lunar month. the period of a complete revolution of the moon around the earth, as the period between successive new moons (synodic month), equal to 29.531 days
http://dictionary.reference.com/browse/synodic+month

The month was originally defined as the time between one new moon to the next, a period now called the synodic month.
http://scienceworld.wolfram.com/astronomy/Month.html

Sometimes other phases of the moon are mentioned but the most common is the new moon.

Synodic month is one moon day

Many students get confused about the geometry of the orbit of the Moon because they have heard this term "the dark side" of the Moon (besides being a famous---at least to us old timers---album by Pink Floyd). If you examine the figures above about how the phases of the Moon change, you see that for at least one half of the lunar orbit, each spot on the Moon has daylight. That is there are 14.75 days when the Sun is visible each orbit, and 14.75 days each orbit when the Sun is not visible. At any one time (just like on the Earth!!!), there is one half of the Moon which is in darkness, and one half which is in the sunlight. It is true that the Moon spins much more slowly than the Earth, but the Moon has "days" and "nights". It is just that these days and nights are 14.75 Earth days long! So the far-side of the Moon which we cannot see is not the "dark side", it is the far-side. At full Moon, the far-side is completely dark. But at New Moon, the far-side is fully illuminated (as seen from the Sun!).
http://ganymede.nmsu.edu/tharriso/ast110/class05.html

What is the point of this quote? We only see one side of the moon. Why? Because the moon is also rotating on its axis. The time the moon has to do “one orbit” about its axis takes the same time as one synodic month. The following picture shows this more clearly:

Earth Moon Day

The 1 Earth day at the beginning is included in the 29.53 days. Note the little yellow spot on the moon. Now as the moon orbits the Earth after one synodic month the little yellow spot moves too. Note the extra moon added for the 3rd Earth position for the synodic month. This is just showing the moon at the opposite side of the Earth. This of course would be half way through the synodic month but for comparison showing it here in the 3rd Earth position. The importance of this is that you can see that the Moon is actually rotating about its own axis as it also rotates about the Earth. Now consider the yellow moon spots on the New Moons. If you were standing on these yellow spots on the moon it would be mid-day. In other words the New Moon to New Moon is mid-day to mid-day for the yellow spot position. What this means is that the New Moon to New Moon has covered exactly one Moon day. What we call the synodic month is no more than the day for the Moon. The synodic month IS one Moon Day. Another way to think of this is that one moon day is equivalent to 29.53 earth days.

Earth rotates approximately 360.986 degrees to complete one earth day
Look at the above picture. The earth yellow spot for mid-day to be in the correct position for the second day must rotate an extra 0.986 degrees. The earth then must rotate 360.986 degrees from mid-day to mid-day. In other words, for one earth day, the earth must rotate approximately 360.986 degrees about its axis.

Moon rotates approximately 389.1 degrees to complete one moon day or synodic month
Look carefully at the Moon positions for the New Moon with the yellow spot marking mid-day to mid-day. That's one Moon Day or synodic month. It's equivalent to 29.53 Earth days. For the Moon, it has had to rotate an extra 29.1 degrees to get there. For one Moon Day or synodic month the Moon has to rotate 360 + 29.1 = 389.1 degrees on its own axis.

Some Bible info on New Moon

חדשׁ
chôdes
h

1) the new moon, month, monthly
1a) the first day of the month
1b) the lunar month
Brown, Driver, and Briggs' Hebrew Definition.

the new moon, the day of the new moon, the calends of a lunar month which was a festival of the ancient Hebrews. (2) a lunar month, beginning at the new moon.
William Gesenius' Hebrew and English Lexicon of the Old Testament. Edited by Francis Brown, S.R.Driver, and Charles A. Briggs.

new moon, month. 1. new moon = day, time, of new moon, as religious festival. 2. month (as beginning with new moon, lunar month; …
Gesenius' Hebrew and Chaldee Lexicon Lexicon to the Old Testament Scriptures.

2320. chodesh, kho'-desh; from 2318; the new moon; by impl. a month:--month(-ly), new moon.
Strong's Exhaustive Concordance of the Bible.

ḥōdesh. Month, monthly, new moon. Although this word properly means “new moon,” it is commonly used as an equivalent to our word “month” because the month began when the thin crescent of the new moon was first visible at sunset. The Hebrew calendar used a lunar month fitted into a solar year. This was done by adding an extra month approximately once every three years because it was about eleven days less than the solar year. In early Israel the first of each month, or new moon, was determined by observation and proclaimed officially by the blowing of trumpets. The month was considered to be thirty days (note Gen 7:11; cf. 8:3-4), unless the new moon was observed earlier.
  When hodesh refers only to the beginning of the month, it is naturally translated “new moon,” which was a feast day. It is one of the “appointed feasts” and is listed with the Sabbath and the pilgrim feasts as involving burnt offerings (II Chr 8:13 et al.), and is also characterized by the blowing of trumpets (Ps 81:3; Num 10:10). Since it was a feast, David's absence from Saul's table at the new moon was especially noticeable (I Sam 20:5f).
Harris, R.L., Archer, Jr., G.L., and Waltke, B.K., Theological Wordbook of the Old Testament, Volume 1, 1981, Moody Press, p. 614.

Most Christian festivals are timed according to the lunar month

lunar calendar, a calendar based on the variations of the phases of the Moon as seen from Earth. The lunar year contains 12 synodic months, these consisting of 29.5305882 days, the synodic month being defined as the time interval (synodic period) between new moons. Therefore, the lunar year of 354.3672 days is about 11 days shorter than the solar year of 365.24219 days. A true lunar calendar quickly gets out of step with the seasons and is often replaced by a luni-solar calendar in which every third or fourth year contains 13 as opposed to 12 lunar months, a leap month being intercalated (added) as required. Islamic, Jewish, Hindu, Buddhist, and all Christian festivals except Christmas itself are timed according to the lunar calendar.
Excerpted from The Oxford Interactive Encyclopedia.1

  Yet just as vital an influence on the calendar is the fact that the Christian Church grew up in the Roman Empire, which followed a calendar controlled by the sun.
  ... when the Church began to institute festivals which were of unique importance for itself and not originally Jewish, they were observed on dates in the Roman calendar. ...
  This is the reason why the Christian calendar is a complicated one. Unlike all others it is a double calendar, governed in some parts by the moon and in others by the sun. The Church distinguishes between 'fixed' and 'movable' feasts, but in reality the movable feasts, such as Easter, are fixed according to the Jewish lunar calendar, while the fixed festivals, such as Christmas, run according to the Roman solar calendar.
  For ten months, from Septuagesima to the last of the Sundays after Pentecost (which vary from twenty-two to twenty-eight), the Christian calendar is lunar. The movable festivals during this period depend upon the date of Easter.2

In 325CE the Council of Nicaea established that Easter would be held on the first Sunday after the first full moon occurring on or after the vernal equinox. From that point forward, the Easter date depended on the ecclesiastical approximation of March 21 for the vernal equinox. Easter is delayed one week if the full moon is on Sunday, which decreases the chances of it falling on the same day as the Jewish Passover. The council’s ruling is contrary to the Quartodecimans, a group of Christians who celebrated Easter on the day of the full moon, 14 days into the month.
http://www.timeanddate.com/calendar/determining-easter-date.html

Moon phases not the same length

Top view rotating moon

This is the view from the Southern Hemisphere. The Moon is rotating clockwise and the line of shadow/light appears to rotate anti-clockwise. From the Southern Hemisphere the line dividing the dark and light parts of the face of the Moon travels from left to right. As θ travels from zero to 2π we see that the full cycle of phases is taken:

θ = 0 New Moon
θ = π/2 First Quarter Moon
θ = π Full Moon
θ = 3π/2 Last Quarter Moon
θ = 2π New Moon again and is the same position for θ = 0.
Note that during this same period of time the Moon has rotated 389.1 degrees about its own axis which is more than 2π radians.

We can determine the formula for x1 in terms of θ mathematically as

x1 = rcos theta

where r is the radius of the Moon. A corresponding velocity formula for x1 as it moves across the face of the Moon can be found as:

velocity formula for x1 across face of moon

The radial rate of change here is different to the radial velocity of the Moon. This is because this calculation is independent of how fast the Moon is rotating itself about its axis. We simply have the phases during the synodic month of 29.53 days. That's it.

As the shadow line gets to the extreme edges [θ approaches 0 and π] the speed will actually drop to zero instantaneously and then increase again with maximum speeds attained at π/2 and 3π/2. These are the radial positions for the First and Last Quarters. The slowest speeds [of zero km/hr] will be attained at the instantaneous points of the New and Full Moons. The mathematics of this gives us an amazing result. The longest phases are the New and Full Moons. All the other phases are shorter!

Formula for percentage of illumination of face of moon
You look at the face of the moon and there is some part in light and the rest dark. What is the percentage of the moon face that is illuminated?
Basically we have something that looks like:

unknown area formula for percentage illumination

The models investigated were rectangular, trapezoidal, parabolic, and exact using spherical coordinate geometry. They all yielded the same formula solution. To simplify the mathematics we will consider the model dimensions about 1 or similar for calculating ratios and only the simplest model is covered here. The rectangular model is here shown:

rectangular model for percentage illumination

The light rectangular area is (1 x1)h , the total rectangular area is 2h. Then the percentage illumination is approximately:

formula for percentage illumination

Jewish new moon not same as astronomical new moon
Apparently the new moon worship day that occurs in the Jewish calendar is not the same as the day of the astronomical new moon [that is, the day when the astronomical new moon occurs].

According to the Mishnah and Tosefta, in the Maccabean, Herodian, and Mishnaic periods, new months were determined by the sighting of a new crescent, with two eye witnesses required to testify to the Sanhedrin to having seen the new lunar crescent at sunset. The practice in the time of Gamaliel II (c. 100 CE) was for witnesses to select the appearance of the moon from a collection of drawings that depicted the crescent in a variety of orientations, only a few of which could be valid in any given month.
http://en.wikipedia.org/wiki/Hebrew_calendar#Importance_of_lunar_months

The molad is the time of the moon's "birth." There is a point in the moon's orbit in which it is positioned directly between the earth and the sun, making it invisible to anyone standing on earth's surface. The molad occurs when the moon has moved far enough from this position that a thin crescent of its illuminated surface becomes visible, marking the start of a new Jewish month.
http://www.chabad.org/library/article_cdo/aid/216238/jewish/Molad-Times.htm#notes


An essential element of the Jewish calendar is the determination of Rosh Chodesh, the day when each month begins.  Since our months, by biblical law, begin with the sighting of the new moon (molad), which can occur on either the thirtieth or thirty-first night of a given month, we need to determine which of these is Rosh Chodesh.  In ancient times, this was done through eyewitnesses, who would testify before the High Court in Jerusalem that they had seen the new moon the previous evening; once they had been cross-examined and judged to be trustworthy, the court would declare the day to be sanctified (Mekudash) as Rosh Chodesh.  This process is known as Kiddush Ha-chodesh (sanctification of the month).  Nowadays, we have a fixed calendar, and we observe either one day of Rosh Chodesh (if the thirtieth day becomes the first of the new month) or two (the thirtieth and the thirty-first, the latter becoming the first day of the new month).  Nevertheless, on the Shabbat preceding Rosh Chodesh (for all months but Tishrei), we make a formal announcement in the synagogue about it, amid prayers for peace and prosperity in the coming month, which we call Birkat Ha-chodesh (blessing of the month).
http://vbm-torah.org/archive/halak68/19halak.htm

Rosh Chodesh, the new month, is marked by the appearance of the first sliver of the new moon in the sky above Jerusalem in Israel.
http://www.bje.org.au/learning/judaism/calendar/index.html

It is clear from these quotations that to the Jewish mind, the new moon occurs when the first sliver of the waxing crescent becomes visible. The following snapshot from a Jewish site shows this clearly:

Jewish site showing Rosh Chodesh new moon

The Jewish new moon is not the same as the astronomical new moon which is completely dark.

Jewish Calendar does not correctly follow the lunar months
The Jewish approach of using Molad dates and times for their calendar construction does not correctly follow the true lunar synodic month through the year. The length of the synodic month changes through the year and the Jewish calendar does not allow for this and some years can actually get a little out of sync.*

The Jews are aware of these length discrepancies and readily admit them:

JUDAISM 101
The Jewish Calendar: A Closer Look
Note that the calculated molad does not necessarily correspond precisely to the astronomical new moon. The length of time from one astronomical new moon to the next varies somewhat because of the eccentric orbits of the Earth and Moon; however, the moladot of Rabbi Hillel's calendar are set using a fixed average length of time: 29 days, 12 hours, and 793 "parts" (or in Hebrew, chalakim). The amount of time is commonly written in an abbreviated form: 29d 12h 793p.
http://www.jewfaq.org/calendr2.htm

Different versions of the new moon day
There are actually quite a few different methods currently being used by various groups around the world for determining the time of the new moon worship day. A rough list would be as follows:

The first sliver of the waxing crescent
The Jewish calendar: similar to waxing crescent but uses a fixed length for the lunar month whereas the moon tends to wander a bit
Conjunction: the exact time to start the new month
Using a numerical value to follow the conjunction and approximate the waxing crescent approach.
Day of the conjunction: similar to conjunction but starts with the day
Day after the astronomical new moon: probably from dawn. And this may be a little different to sunrise
Beginning of the dark phase of the moon
The full moon. Yes you have read right. Some believe that the current full moon is the true new moon of the Bible and has been lost in time

and this list may not be exhaustive!

Various reasons for keeping the new moon day
As with the previous list of different methods currently being used by various groups around the world for determining the time of the new moon worship day there are also various reasons for keeping the day. A rough list would be as follows:

A holiday for women.

A worship day and kept strictly as a sabbath rest: no business or work activities of any type to be undertaken.

A worship day for giving thanks to the Lord for all the joy and happiness that He is giving us each lunar month. Not a sabbath rest. Work can be done on the day.

A worship day for the new moon festival of humility. Ceremony of footwashing and is usually done before partaking of the Lord's supper. Probably a sabbath rest. No work activities.

A rest day from certain activities but not necessarily work. Some yoga groups.

and this list may not be exhaustive!

A puzzle
Finally, we will leave our athiest friends with two quotations to puzzle over: one from the Bible and the other from some website that may not have any Christian leaning. Enjoy.

Gen 1:16 And God made the two great lights, the greater light to rule the day, and the lesser light to rule the night; he made the stars also.

The Moon orbits the Earth - or more accurately the Earth and Moon revolve around a common barycentre - at a centre-to-centre distance of 384,000 km. The Sun, some 150,000,000 km away, is 1,400,000 km across. It is an almost unbelievable coincidence that from the surface of the Earth, the Sun and the Moon appear more or less exactly the same size - around 32 arc minutes. It is this fact which allows us to see the Sun’s corona during a solar eclipse. No other combination of bodies anywhere in the Solar System give rise to this phenomenon.
http://www.inconstantmoon.com/not_dime.htm


* It got completely out of step in 2013. This was one of those out-of-sync years. See the PostScript section at the end of my Search page.

Isaiah 66:22,23 quoted

1. The Oxford Interactive Encyclopedia. Developed by The Learning Company, Inc. Copyright (c) 1997 TLC Properties Inc. All rights reserved.
2. L.W.Cowie and John Selwyn Gummer, The Christian Calendar: A complete guide to the seasons of the Christian year, 1974, Weidenfeld and Nicolson, London, pp. 7,8.
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Stephen Buckley
E-mail: stephen@chodesh.info
Last revised: 18 Sep 2016.