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Higher Arithmetic as a Machine Tool:
Gauss' Determination of the Easter Date
Last week's pedagogical discussion ended with the
provocative question: "If there exists no grand mathematical
system which can combine and account for the various cycles, then
how can we conceptualize the `One' which subsumes the successive
emergence of new astronomical cycles as apparent new degrees of
freedom of action in our Universe? How do we master the
paradoxical principle of Heraclitus, that `nothing is constant
except change?'"
This problem was attacked in a very simple and beautiful way
by C.F. Gauss, using purely the principles of higher arithmetic,
in his determination of the Easter date. Since the last
conference presentation, I have received several requests, to
elaborate more completely the derivation of Gauss' algorithm.
While the development of Gauss' program requires no special
mathematical skills other than simple arithmetic, it does require
the conceptual skills of higher arithmetic, i.e., the ability for
the mind to unify an increasingly complex Many into a One. This
is a subjective question. We are not looking for one
mathematical formula, but a series of actions, which, when
undertaken, enable our minds to wrestle a seemingly unwieldy
collection of incommensurable cycles into our conceptual grasp.
In a certain sense, we are designing and building a machine tool
to do the job, but only the entire machine can accomplish the
task. No single part, or collection of parts, will be
sufficient. The whole machine includes not only the "moving
parts," but the concepts behind those moving parts. All this,
the parts and the concepts, must be thought of as a "One," or
else, the machine, i.e., your own mind, comes to screeching halt,
while the earth, the moon, the sun, and the stars, continue their
motion, in complete defiance of your blocking.
Over the next few weeks, we will re-discover Gauss'
construction. But, in order to build this machine, you must be
willing to get your hands dirty and break a sweat, make careful
designs, cut the parts to precision, lift heavy components into
place, and finally apply the energy (agape) necessary to get the
machine moving and keep it moving.
In the beginning of his essay, "Calculation of Easter,"
published in the August 1800 edition of Freiherrn v. Zach's
"Monthly Correspondence for the Promotion of News of the Earth
and Heavens," Gauss states:
"The purpose of this essay, is not to discuss the usual
procedure to determine the Easter date, that one finds in every
course on mathematical chronology, and as such, is easy enough to
satisfy, if one knows the meaning and use of the customary terms
of art, such as Golden Number, Epact, Easter Moon, Solar Cycle,
and Sunday Letter, and has the necessary helping tables; but this
task is to give, independently from those helping conceptions, a
purely analytical solution based on merely the simplest
calculation-operations. I hope, this will not be disagreeable,
not only to the mere enthusiast who is not familiar with those
methods, or for the case where one wishes to determine the Easter
date, under conditions in which the necessary helping devices are
not at hand, or for a year which cannot be looked up in a
calendar; but it also recommends itself to the expert by its
simplicity and flexibility."
This article was published after Gauss had completed, and
was awaiting publication of the "Disquistiones Arithmeticae."
Of the principles we will develop here, Gauss says:
"The analysis, by means of which the above formulas are
founded, is based properly on the foundations of {Higher
Arithmetic}, in consideration of which I can refer presently to
nothing written, and for that reason it cannot be freely
presented here in its complete simplicity: in the mean time, the
following will be sufficient, in order to lay the foundation of
the direction of the concept and to convince you of its
correctness."
Gauss' choice of the problem of determining the Easter date,
to demonstrate the validity of the principles of his Higher
Arithmetic, is not without a healthy amount of irony, but the
resulting calculation was by no means Gauss' only goal. As with
LaRouche's current program of pedagogical exercises, Gauss
recognized the effectiveness for increasing the conceptual powers
of the human mind, of working through specific examples, which
demonstrate matters of principle. Gauss continued this approach
in all his work, demonstrating new principles as he conquered one
problem after another. Gauss repeatedly found that in these
matters of principle, connections were discovered between areas
of knowledge which were previously thought to be unrelated.
From the earliest cultures, the various cycles described
last week were accounted for separately, and their juxtaposition
was studied with aid of the different tables and calculations
Gauss mentioned above. These methods were adequate for
determining the date of Easter from year to year. Gauss'
calculation is purely a demonstration of the power of the human
mind, to create a new mathematics, capable of bringing into a
"One" that which the previous state of knowledge considered
unintelligible. For that reason, it suits our present purpose.
To begin, we should think about the problem we intend to
work through: To determine the date of Easter for any year.
Easter occurs on the first Sunday, after the first full Moon
(called the Paschal Moon) after the Vernal Equinox. This entails
three incommensurable astronomical cycles: the day, the solar
year, and the lunar month; and one socially-determined cycle, the
seven-day week.
Now look more closely at what this "machine-tool" must do:
1. It must determine the number of days after the vernal
equinox, on which the Paschal Moon occurs. This changes from
year to year. So the machine must have a function, which
modulates the solar year (365.24 days) with the lunar month
(29.53 days).
2. Once this is determined the machine must also determine
the number of days, remaining until the next Sunday.
The incommensurability of the solar year and the lunar month
is an ancient conceptual problem, upon whose resolution man's
potential for economic progress rested. If one relied solely on
the easier-to-see lunar month, the seasons (which result from
changes of the position of the earth with respect to the sun)
will occur at different times of the year, from one year to the
next. On the other hand, if one relies on the solar year, some
intermediate division between the day and year is necessary, to
measure smaller intervals of time. Efforts to combine both the
lunar cycle, and the solar cycle, linearly into one calendar,
creates a complicated mess. The Babylonian-influenced Hebrew
calendar is an example, requiring a special priestly knowledge
just to read the calendar. Shortly after the publication of the
Easter formula, Gauss applied the same method to a much more
complex chronological problem, the determination of the first day
of Passover, and in so doing, subjugating the Babylonian
lunisolar calendar to the powers of Higher Arithmetic.
In 423 B.C., the Greek astronomer Meton reportedly
discovered that 19 solar years contained 235 lunar months. This
is the smallest number of solar years, that contain an integral
number of lunar months. There is evidence that other cultures,
including the Chinese, discovered this same congruence earlier.
By the following simple calculation, we can re-discover Meton's
discovery. One solar year is 365.2425 days. 12 lunar months is
354.36 days, (12 x 29.53) or 11 days less than the solar year.
This means that each phase of the moon will occur 11 days earlier
than the year before, when compared to the solar calendar.
(For example, if the new moon falls on January 1, then after
12 lunar months, a new moon will fall on December 20 -- 11 days
before the next January 1. The next new moon will occur on
January 19, 19 days after the next January 1.)
One solar year contains 12.368 lunar months (365.2425 /
29.530). In 19 years, there are 6939.6075 days (365.2425 x 19).
In 19 years of 12.368 lunar months, there are 6939.3137 days (19
x 12.368 x 29.530). That is, if you take a cycle of 6939 days,
or 19 solar years, the phases of the moon and the days of the
solar year become congruent.
Despite Meton's discovery, the Greek calendar was still
encumbered by a failed effort to combine the lunar months and
solar year into a single linear calendar cycle. Since 12 lunar
months, are 11 days short of the solar year, the Metonic
calendar, like the Babylonian influenced Hebrew calendar,
required the intercalation (insertion) of leap months in years 3,
5, 8, 11, 13, and 16 of the 19-year cycle.
In his "History," Herodotus remarks on the inferiority of
the Greek method over the Egyptians, whose calendar was based
only on the harder-to-measure solar year. "But as to human
affairs, this was the account in which they all agreed: the
Egyptians, they said, were the first men who reckoned by years
and made the year consist of twelve divisions of the seasons.
They discovered this from the stars (so they said). And their
reckoning is, to my mind, a juster one than that of the Greeks;
for the Greeks add an intercalary month every other year, so that
the seasons agree; but the Egyptians, reckoning thirty days to
each of the twelve months, add five days in every year over and
above the total, and thus the completed circle of seasons is made
to agree with the calendar."
The oligarchical view of this matter is expressed by the
Chorus-Leader in Aristophanes, "The Clouds":
"As we prepared to set off on our journey here,
The Moon by chance ran into us and said she wanted
To say hello to all the Athenians and their allies,
but she's most annoyed at your treating her so shamefully
despite her many evident and actual benefactions.
First off, she saves you at least ten drachmas a month in
torches:
that's why you all can say, when you go out in the evening,
No need to buy a torch, my boy, the moonlight's fine!
She says she helps in other ways too. But you don't keep
your calendar correct; it's totally out of sync.
As a result, the gods are always getting mad at her,
whenever they miss a dinner and hungrily go home
because you're celebrating their festival on the wrong day,
or hearing legal cases or torturing slaves instead of
sacrificing.
And often, when we gods are mounring Memnon or Sarpedon,
you're pouring wine and laughing. That's why Hyperbolus,
this year's sacred ambassador, had his wreath of office
blown off his head by us gods, so that he'll remember well
that the days of your lives should be reckoned by the Moon."
In 46 B.C., with the adoption of the Julian calendar, all
attempts to incorporate the lunar cycle into the calendar were
abandoned. But, it wasn't until Gauss' development of higher
arithmetic, ironically based on a re-working and non-linear
extension of classical Greek astronomy and geometry, that man had
the ability to encompass the seemingly incommensurable lunar
month and solar year into a One.
With these discoveries in mind, we can begin to construct
the first components of the machine, which will determine the
number of days from the vernal equinox, to the Paschal Moon. If
we fix the vernal equinox at March 21, our first component must
determine some number D, which, when added to March 21, will be
the date of the Paschal Moon. (March 21 was the date set at the
Council of Nicea. The actual Vernal Equinox, can sometimes occur
in the late hours of March 20, or the early hours of March 22.)
The Paschal Moon will occur on one of 30 days, the earliest being
March 21, the latest being April 19. The variation from year to
year, among these 30 possible days, is a reflection of the 19-
year Metonic cycle. So, our machine, must make two cycles, the
19-year Metonic cycle, and this 30-day cycle into a One.
This requires some thinking. Since 12 lunar months are 11
days less than the solar year, any particular full moon will
occur 11 days earlier than the year before. Naive imagination
tells us that if we set our machine on any given year, all it
need do is subtract 11 days to find the Paschal Moon on the next
year. But we have a boundary condition to contend with. The
Paschal Moon can never occur before March 21. So, when the
Paschal Moon occurs in March, and our machine subtracts 11 days,
to get the date of the Paschal Moon the following year, the new
date will be before March 21. That will do us no good at all.
To determine the date of the Paschal Moon from one year to
the next, our machine must do something different when the
Paschal Moon occurs in March, than when it occurs in April. When
the Paschal Moon occurs in April, the machine must subtract 11
days, to determine the date for the following year. But when it
occurs in March, the machine must add 19 days to determine the
date for the following year.
To construct this component of the algorithm, Gauss began
with a known date, and abstracted the year-to-year changes, with
respect to that date. In reference to the 19-year Metonic cycle,
he chose to begin the calculation with the date of the Paschal
Moon in the first year of that cycle (i.e., those years which,
when divided by 19, leave 0 as a remainder, or are congruent to 0
relative to modulus 19). In the 18th and 19th centuries, that
date was April 13, or March 21 + 23 days.
For clarity, we can make the following chart:
Year Residue Paschal Moon # Days Aft. Equinox (D)
(Mod 19)
1710 0 April 13 23 days
1711 1 April 2 23 - 11 days
1712 2 March 22 23 - (2 x 11)
1713 3 April 10 23 - (2 x 11) + 19
1714 4 March 30 23 - (3 x 11) + 19
(The reader is encouraged to complete this entire chart. When you
do this notice the interplay between the 19 year, and 30 day
cycles.)
From the chart, you should be able to see the relevant
oscillation. For example, for year 1713, were we to have
subtracted another 11 days from the year before, we'd arrive at
the date of March 11. A full moon certainly occurred on that
day, but it wasn't the Paschal Moon, because March 11 is before
the Vernal Equinox. The Paschal Moon, in the year 1713, occurred
30 days later than March 11, on April 10. (March 22 - 11 + 30;
or March 22 + 19)
The number of days added or subtracted changes from year to
year, in a seemingly non-regular way. What is constant is
change. But this step-by-step process, is really no different
than if we had a series of tables.
Gauss' next step, is to transform the two actions,
subtracting 11 days or adding 19 days, into one action. There
are many ways this can be done. The determination of the
appropriate one, is a matter of analysis situs, and involves one
of the most important methods of scientific inquiry: {inversion}.
The principle of inversion is characteristic of all Gauss' work.
It is one thing to be given a function, and then calculate the
result. The inverse question is much more difficult. Given a
result, what are the conditions which brought about that result?
In the latter case, there are many possible such conditions,
which cannot be ordered without consideration of higher
dimensionalities. (This subject will be treated more in future
pedagogical discussions.)
Our immediate problem can be solved, if we think about it
from the standpoint of inversion. All the year-to-year
differences between the dates of the Paschal Moon, are either
congruent relative to modulus 11 or modulus 19. But neither of
these moduli are relevant for the task at hand. A different
modulus must be discovered, which is not self-evident from the
chart, but is evident from the higher dimensionality of the
complete process. As discovered earlier, the Paschal Moon occurs
on one of 30 days between March 21, and April 19. We need to
discover a means, under which the oscillation of the dates of the
Paschal Moon, can be ordered with respect to modulus 30. If we
number these days 0-29, the numbers 0 to 29 each represent
different days, and are all non-congruent relative to modulus 30.
Gauss chose to combine the two actions into one, by adding
19 days to {every} year, and subtracting 30 days from those years
in which the Paschal Moon occurs in April. (For example, in our
chart above, the year 1711 would be calculated: 23 + 19 - 30; the
year 1712 would be calculated, 23 + (2 x 19) - (2 x 30).
Since all numbers whose differences are divisible by 30, are
all congruent relative to modulus 30, adding or subtracting 30
days from any interval, will not change the result. Gauss has
transformed this problem into a congruence relative to a single
modulus: 30. So the first component of our machine takes the
year, finds the residue, multiplies that by 19, adds 23, divides
by 30 and the remainder is the number of days from the Vernal
Equinox to the Paschal Moon.
Or in Gauss' more condensed language: Divide the year by 19
and call the remainder a. Then divide (23 + 19a) by 30 and call
the remainder D. Add D to March 21 to get the date of the
Paschal Moon.
No mountain was ever climbed that didn't require some sweat.
Or, put another way, in order to build the Landbridge, you have
to move some dirt.
Next week: From the Paschal Moon to Easter.
Higher Arithmetic as a Machine Tool--Part II:
Gauss' Determination of the Easter Date
Last week we completed the first step of the development of
Gauss' algorithm for calculating the Easter date, using the
principles of Higher Arithmetic. This week we continue the climb.
Those experienced in climbing mountains are aware, that as one
approaches the peak, the climb often steepens, requiring the
climber to find a second burst of energy. Even though last week's
climb might have required some exertion, you've had a week's
rest, and a national conference in the intervening period. Armed
with the higher conceptions of man expressed by Lyn and Helga at
the conference, everyone is well-equipped to complete this climb.
Again it is important to keep in mind, that the
determination of the date of Easter was not a goal in itself for
Gauss. Rather, Gauss understood that working through problems,
which required the discovery of new principles, was the only way
to advance human knowledge.
Last week, we worked through the first part of the task of
determining the date of Easter. Since Easter is the first Sunday
after the first full moon, after the vernal equinox, the first
job of our machine tool, is to determine the date of the first
full moon. This requires bringing into a One, three astronomical
cycles: the day, the lunar month, and the solar year. The second
part of the job, to determine the number of days from the Paschal
Moon until the next Sunday, requires bringing into a One, various
imperfect states of human knowledge.
It was a major step forward, for society to abandon all
attempts to reconcile the lunar and solar years into one linear
calendar, and adopt the solar year, as the primary cycle on which
the calendar was based. The conceptual leap involved was to base
the calendar on the more difficult to determine solar year,
instead of the easier to see lunar months. The implications of
this conceptual leap for physical economy are obvious. What is
worth emphasizing here, is, that this is a purely subjective
matter, whose resolution determines physical processes. This
development, however, was not without its own problems.
While the disaster of trying to reconcile the lunar and
solar cycles, becomes evident within the span of several years,
the problems of the solar calendar, don't become significant
within in the span of a single human life.
As discussed last week, the solar year is approximately
365.24 days. In 46 B.C., the calendar reform under Roman Emperor
Julius Caesar, set the solar year at 365.25 days, which was
reflected in the calendar, by three years of 365 days, followed
by a leap year of 366 days. The number of days in this
arrangement, would coincide every four years. Under this
arrangement, man has imposed on the astronomical cycles, a new
four-year cycle. From the standpoint of Gauss' Higher
Arithmetic, leap years are congruent, in succession to 0 relative
to modulus 4, followed by non-leap years congruent to 1, 2, or 3
relative to modulus 4.
Like all oligarchs who delude themselves that their rule
will last forever, Julius Caesar's arrogance of ignoring the
approximately .01 discrepancy between his year, and the actual
astronomical cycle, became evident long after his Empire had been
destroyed. This .01 discrepancy, while infinitesimal with
respect to a single human life, becomes significant with respect
to generations, causing the year to fall one day behind every 187
Julian years. By the mid-16th century, this discrepancy had
grown to 11 days, so the astronomical event known as the vernal
equinox was occurring on March 10th instead of March 21st. The
economic implications of such a discrepancy is obvious.
This lead to the calendar reforms of Pope Gregory XIII in
1587. In the Gregorian calendar, the leap year is dropped every
century year, except those century years divisible by 400. This
decreases the discrepancy of the .01 day, but doesn't eliminate
it altogether. In order to get the years back into synch with the
seasons, Pope Gregory dropped 11 days from the year 1587. Other
countries reformed their calendar much later, having to drop more
days, the longer they waited. The Protestant states of Germany,
where Gauss lived, didn't adopt the calendar reform until the
early 1700s. The English didn't change their calendar until
1752. The Russians waited until the Bolshevik revolution.
The other human cycle involved in this next step of the
problem is the seven-day week. There is no astronomical cycle
which corresponds to the seven-day week. While the Old
Testament's Exodus, attributes the seven-day week to God's
creation of the universe, Philo of Alexandria, in his
commentaries on the Creation, cautions that this cannot be taken
literally. Philo says the Creation story in Genesis 1, must be
thought of as an ordering principle, not a literal time-table.
Here is another example of what Lyn has discussed about the
unreliability of a literal reading of the Old Testament. The
idea that creation took seven days, shows up in Exodus,
contradicting the conception of an ordering principle of Creation
in Genesis 1.
Of importance for our present problem, is that, the
seven-day weekly cycle runs continuously, and independently, from
the cycles of the months, (either calendar or lunar) and the
years. What emerges is a new cycle which has to be accounted for.
Each year, the days of the week occur on different dates. For
example, if today is Saturday, September 6, next year, September
6 will be on a Sunday. However, when a leap year intervenes, the
calendar dates move up two days. This interplay between the
seven-day week and the leap year, creates a 28-year cycle, before
the days of the week and the calendar dates coincide again. This
cycle also has to be accounted for in Gauss' algorithm.
So, to climb that last step, from the Paschal Moon to
Easter, we have to bring into a One, these two human cycles, the
leap year, and the seven-day week.
Before going any further, one must first remember a
principle of Higher Arithmetic. Under Gauss' conception of
congruence, it is the {interval} between the numbers, on which
the congruence is based, not the numbers themselves. We are
relating numbers by their intervals. Consequently, when we add
or subtract multiples of the modulus to any given number, the
congruence relative to that modulus doesn't change. For example,
15 is congruent to 1,926 relative to modulus 7. The interval
between 15 and 1,926 (1,911) is divisible by 7. If, for example,
we subtract 371 (7x53) from 1,926, the result will still be
congruent to 15. The reader should do several experiments with
this concept, in preparation for what follows.
It were useful to restate here Gauss' entire algorithm:
Divide the year by 19 and call the remainder a
Divide the year by 4 and call the remainder b
Divide the year by 7 and call the remainder c
Divide 19a+23 by 30 and call the remainder d
(This was discovered last week)
Divide 2b+4c+6d+3 by 7 and call the remainder e
(This is today's task.)
The number of days from the Paschal Moon until Easter Sunday
can be at least 1 and at most 7 days. Because Easter is the
first Sunday {after} the first full moon, which follows the
Vernal Equinox, the earliest possible date for Easter is March
22. Therefore, Easter will fall on March 22 + d (the number of
days to the Paschal Moon) + E (the number of days until Sunday.)
E, therefore, will be one of the numbers 0-6, or the least
positive residues of modulus 7.
Keeping in mind the exercise we discussed above, the number
of days between any two Sundays is always divisible by 7, no
matter how many weeks intervene. Consequently, the interval of
time between March 22 + d + E (Easter Sunday of the year we're
trying to determine) and any given Sunday in any previous year,
will be divisible by 7. So if we begin with a definite Sunday,
we can discover a general relationship for determining the date
of Easter.
Gauss chose Sunday, March 21, 1700 as his Sunday reference
date. Next, he determined a relationship for how many total days
elapsed between March 21, 1700 and any subsequent Easter Sunday.
That total would be 365 days times the number of elapsed years,
plus the number of leap days in those elapsed years. (Remember
every four years, has one leap day in it.) Again, this number
will be divisible by 7, no matter how many years intervene.
If A is the year for which we want to determine the date of
Easter, A-1700 will be the number of elapsed years. (For
example, if we want to find Easter in the year 1787, then there
were 87 elapsed years (1787-1700).
If we call i the total number of leap days, then the total
number of days between Sunday March 21, 1700 and March 22 + d +
E, for the year we're investigating will be: