__Calendars__

__Mayan religion__: the *ceiba* tree a central metaphor as a tree of life. It existed in an underworld (roots) in
the earth (trunk) and in the heavens (branches). It exhaled the breath of Hunab KÕU, the creative force. Without trees, no life. Human sacrifice was practiced to
appease hungry gods, though probably not on Aztec scales (sometimes ca. 1000s
in one festival for the Aztecs).

__Vigesimal
Number System__:
The Maya also developed perhaps the most sophisticated number system before the
decimal (base-10) system we use now (originating from the 15^{th}
century and utilizing Hindu-Arabic symbols and place-holding zeros). The Mayan system is based on 20s
(vigesimal = 20-based) and also used zeros to hold places and denote the
somewhat abstract idea of nothing.
This numerical concept was not usable in the Roman numeral system (e.g.
VII or MCMLIX), although the idea of nothing was certainly in the Latin
language (*nihil* or *nil*). It is often stated that the concept of ÒzeroÓ was unknown to
the Greeks and Romans and Medieval Europeans. This is inaccurate.
The number ÒzeroÓ generally did not exist in their computational
systems, but the concept was clearly available to them. [Ptolemy, an Egyptian from the 2^{nd}
c. AD in Roman–ruled Egypt, actually used a zero in his hybrid
computational system, but this is generally disregarded by historians who
usually donÕt understand enough math to say anything coherent. This is a good example of historians
not knowing what they are talking about.]
None-the-less, the Maya had a zero, and it was utilized in their general
system and was not just an anomalous occurrence.

Example of vigesimal numbers
compared to our decimal system.

Decimal |
Decimal
mimicking Vigesimal style |
Decimal = Vigesimal |
Vigesimal
(std. notation) |

1
= 1 x 10 |
1 |
1
= 1 x 20 |
1 |

3
= 3 x 10 |
3 |
3
= 3 x 20 |
3 |

10
= 1 x 10 |
1|0 |
10
= 10 x 20 |
10 |

20
= 2 x 10 |
2|0 |
20
= 1 x 20 |
1|0 |

30
= 3 x 10 |
3|0 |
30
= (1 x 20 |
1|10 |

100
= 1 x 10 |
1|0|0 |
100
= 5 x 20 |
5|0 |

125
= (1 x 10 |
1|2|5 |
125
= (6 x 20 |
6|5 |

135
= (1 x 10 |
1|3|5 |
135
= (6 x 20 |
6|15 |

This
makes computation much, much easier to do on paper. All you have to do is add or subtract corresponding
powers.

Decimal |
Vigesimal
(same idea) |

135
= 031 = 31 |
6|15
=
1|11 (= 31
in decimal notation) |

Notes from MannÕs book (pp. 403-407) are in blue.

__Mayan Calendar__: The Mayan Calendar was arguably the most sophisticated
calendrical system produced before the modern age.

The
Maya had two time divisions (three if you count the combined Long Count) of
note:

-One was 260 days long, called *Tzolk'in.* It is in evidence as early as 200 BC. It was divided into 13 20-day
Òmonths.Ó It is 260 days long
because this is the lowest common multiple of 13 and 20.

Mann describes this somewhat differently, calling the 13-day
cycle the month because it is numbered, and the 20-day cycle as a week because
its days are named like we name Tuesday and Wednesday. This system produces a
unique combination which only repeats after 260-day cycle has been completed.

Me: example: first
place is base 13 and the second place is base 20. First day of the year would be 1.1 and the last would be
13.20. If our months were
consistently 30 days long, our system would cycle every 210 days.

The number 20, as already noted,
is the basis for the Mayan number system and most literature likes to suggest
that this number may have developed from the number of fingers and toes. This is sheer speculation, but not
unreasonable speculation in my estimation. But it does make you wonder if shoe wearing is a significant
factor in the development of number systems?

The number 13 is the number of
layers in the Mayan heaven.

Why
260?

PerhapsÉ260 closely approximates
the number of days in a typical pregnancy (calculated to be 266) and is roughly
the number of days in a typical southern Mexican growing season.

Mann speculated that
the visible times of Venus as morning star, may have inspired this system. ItÕs visible in the morning for 263
days, then behind the sun for 50, then visible as the evening star for 263 days.

-The other was 365-days long
(approximately a __solar__ year) (called *Haab'*)
and was divided into 18 20-day ÒmonthsÓ with 5 additional days considered to be
unlucky tacked on at the end. When performing calendrical calculations, the
Mayan number system was modified to incorporate this 18-ÒmonthÓ year. Instead of the strict vigesimal
(base-20) system, the 20^{2} place (20 x 20) was changed to 18 x
20. Thus one dot in this part of a
number represented 360 [1 x 18 x 20], two dots equaled 720 [2 x 18 x 20], and
so onÉ This is similar to our way of writing dates as 10/26/10. Each place is not based on 10, but is based on the number of
months, number of days, and number of years. Try adding in this whacky system like we add our decimal
number system or the vigesimal system.
Not very easy. Of course
adding or subtracting dates in our system is not so easy either. You cannot simply add 15 days [0/15/0]
to 10/26/2010. You'd get
10/41/2010 and then have to convert the "41" to the next month with a
remainder of days, taking into account that not all months have an equal number
of days. It's messy.

From Mann, p. 405.

The Long Count: Mann describes this quite differently.

The combination of these two
cycles (the *Tzolk'in *and the *HaabÕ*) results in what is referred to as
the ÒLong Count.Ó If both systems
start at the same time, it will take 52 *HaabÕ*
(years) for both systems to synch up again. It will take 73 *Tzolk'in
(73 x 260 days).*

52 x
365 days = 18,980 days

73 x
260 days= 18,980 days

This cycle was of considerable
importance. It is
coincidentally(?) about the lifespan of a long-lived Mayan (ca. 52 years).

As you can see, none of these
systems include leap years of any sort, so this system simply accumulated
ÒerrorÓ over the centuries and it cycled around completely every 1460
years.

The sophistication of the Mayan
calendar has captured the imagination of historians for many years. The Mayan solar year has been
calculated by some to be approximately 365.2422 days long, although the .2422
was not incorporated into the calendar.
(One should treat this number with much skepticism, but IÕll accept it
for fun.) The modern value is generally given as approx. 365.24219 for what is
referred to as the mean tropical year or approx. 365.2424 days for the vernal
equinox year. These are just
slightly different ways to measure the solar year. The difference between these
years is measured by the 1/10,000 of a day. Even if we use the vernal equinox
year which differs from the Mayan year more significantly, the difference
between the two years is only 18 seconds. It would take 200 years for the two
systems to differ by an hour, and 4800 years to differ by a full day. But it should be noted that the Maya
did not treat fractional days in any way and their calendars did not reflect
this level of accuracy in the way that we have leap years and leap seconds and
the like, even though such accuracy appears to be inferred from their
astronomical observations.

*b'ak'tun
= 144,000 days *(18 X 20^{3})

There
is yet another system of time lasting approximately 394 years called *b'ak'tun, *a word which more or less
refers to the calendrical number system which differed from the regular number
system in that instead of the 20^{2} place it used 18 x 20 (referred to
above). This 360-day grouping was
called a *tun (18 x 20 ^{1})*. The next vigesimal place was called the

Using
this system, the Maya established a beginning year: ca. 3114 BC. (Compare with the beginning for the
Jewish calendar: 3760 BC.)
Although there are references to events before this date in some
inscriptions. It was prophesized
in some writings that at the __13 ^{th} b'ak'tun__ the world would enter a new cycle of existence, whatever
that means. This new cycle
corresponds to our year 2012 AD.
Did you see the movie?

The Maya wrote dates like so: # *b'ak'tun
– # ?, #?, # ??? É.*Mann
is horribly vague. But they wrote
the dates in 5 places from greatest to least division. IÕm thinking: [X x 18 x 20^{3}], [Y x 18 x 20^{2}],
[Z x 18 x 20], [# of 20 day cycle], [day in 365 system]

Me: the system seems to be based on this: 18 X 20^{n}.

An even longer division than the *bÕakÕtun* was called the *alautun*,
which is 23,040,000,000 days long (ca. 63 million years). [18 X 20^{7}]

The counts in order of duration:

13 days, 20 days, 260 days, 365
days, 7,200 days, 18,980 days (= 52 years. the lowest common multiple of 260 and
365), 144,000 days (ca. 394 years).

Similar
to the Maya, the Aztecs believed that a series of cosmic cycles that preceded
the present time.

This is about
12 feet in diameter and is in Mexico City.

This
shows the previous 4 ages of the world.

The first age (upper right
quadrant) tells of giants created by the god. But they did not till the earth (no agriculture) and the
gods sent jaguars to eat them.

The second age (upper left
quadrant) tells of another race created by the gods that didnÕt work out. They became apes and escape a windy
destruction by clinging to the world tightly.

The third age (lower left
quadrant) tells of another creation resulting in birds who escape a volcanic
apocalypse.

The fourth age (lower right
quadrant) tells of another aborted creation which resulted in fish who escaped
a cataclysmic flood.

In the next age (presumable the
5^{th} epoch) we are what the gods created and we must work to appease
the gods who will otherwise destroy the world with an earthquake.

Because the gods made sacrifices
in order that we may exist, it stands to reason that we should return the favor
and sacrifice to the gods. Hence,
human sacrifice.

Stonehenge
– England A view of the
Summer Solstice.

-Built in 3 phases spanning
3,100 to 1500 BC

-Think of it as a large clock.

-Summer solstice (June 21)
sunrises exactly from the ÒHeel StoneÓ (see above)

-Winter Solstice and the two
equinoxes are also indicated by stones as well as other aspects of lunar
motions.

__Gregorian Calendar: Implamented
by Pope Gregory XIII in 1582__

Leap
years are all years divisible by 4, with the exception of those divisible by
100, but not by 400. These 366-day years add a 29th day to February, which
normally has 28 days. (So, in the last millennium, 1600 and 2000 were leap
years, but 1700, 1800 and 1900 were not. In this millennium, 2100, 2200, 2300
and 2500 will not be leap years, but 2400 will be.)

Easter
established at first council of Nicaea in 325.

The
problem was, people used a calendar to tell them when the vernal equinox was,
rather than observation.

A new
calendrical system was developed to replace the Julian system which was simply
not working. When the new calendar was put in use, the error accumulated in the
13 centuries since the Council of Nicaea was corrected by a deletion of ten
days. The last day of the Julian calendar was Thursday 4 October 1582 and this
was followed by the first day of the Gregorian calendar, Friday 15 October 1582
(the cycle of weekdays was not affected). Nevertheless, the dates "5
October 1582" to "14 October 1582" (inclusive) are still valid
in virtually all countries because even most Roman Catholic countries did not
adopt the new calendar on the date specified by the bull, but months or even
years later (the last in 1587).
Non-Catholic countries like England resisted this change but eventually
gave in to its functionality.

Alaska
adopted it in 1867 and Greece in 1923, China in 1912, Japan in 1873,