Higher Arithmetic as a Machine Tool

by Bruce Director

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

by Bruce Director

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 (7×53) 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:

1 + d + E + i + 365(A-1700)

This number is divisible by 7, (because it is the number of intervening days from one Sunday to another).

At this point, the main conceptual problem has been solved. The date of Easter can be determined as March 22 + d + E, with d being determined by the calculation discussed last week, and E determined by the calculation which will be developed below.

Gauss was never content, unless he found the absolute simplest way to accomplish his task. All that remains is to simplify the above calculation so that E will be the residue which arises when the above number is divided by 7. Gauss accomplished this by repeatedly employing the principle, cited above, that adding or subtracting multiples of the modulus, doesn’t change the congruence. I include the following applications of this principle, even though it is expressed by some algebraic manipulations. The reader should focus on the addition and subtraction of multiples of the modulus 7.

To determine the number of leap days, i. we must first determine what relationship the year in question is to the leap year. Or, in the language of Gauss’ Higher Arithmetic, what is the least positive residue relative to modulus 4 of the year in question? This is the remainder b in Gauss’ algorithm. (For example, if the year is 1787, the least positive residue relative to modulus 4 is 3. That is, 1787 is three years after a leap year. So the total number of leap years between 1787 and 1700 is 87-3/4=21, or 1787-1700-3/4.)

In Gauss’ formula, the total number of leap days i will be:


If A is between 1700 or 1799. If A is between 1800 and 1899, then we have to subtract 1 because 1800 is not a leap year. For now, we will stick to the 18th century.

So the total number of days between March 21, 1700 and Easter Sunday in year A, will be:

1 + d + E + 365(A-1700) + 1/4(A-b-1700).

And this number must be divisible by 7.

This is pretty complicated and cumbersome. But as we know from Gauss’ Higher Arithmetic, if we add or subtract multiples of 7, the result will also be divisible by 7. So Gauss, through the following steps, adds or subtracts multiples of 7, in order to bring this unwieldy formula into a simple calculation.

First he adds the fraction 7/4(A-b-1700) to the above making: 1 + d + E + 365(A-1700) + 8/4(A-b-1700)

Multiplying all this out gives us: 1 + d + E + 367(A-1700)-2b which equals: 1 + d + E + 367A – 623,900 – 2b

Then Gauss subtracts 364(A-1700) (which is divisible by 7) which gives: d + E + 3A – 5099 – 2b

Then Gauss adds 5096 (which is divisible by 7) to get: d + E + 3A – 3 – 2b

Now Gauss eliminates any need for the reference date by replacing A in the following way. First, we divide the year by 7 and call the remainder c. That means, if we subtract c from the year, the result will be divisible by 7. Or, A-c will also be divisible by 7. In the next step, Gauss subtracts 3 times A-c or 3A-3c which gives: d + E + 3c – 3 – 2b

Finally Gauss subtracts this from 7c – 7d which gives: 3 + 2b + 4c + 6d – E.

Which means E is the remainder if we divide 3 + 2b + 4c + 6d by 7. So the determination of the Easter date is March 22 + d + E.

Unfortunately our work is not completely done. Because in the Gregorian calendar, not every century year is a leap year, the algorithm must change from century to century. Gauss also solved this problem using principles of Higher Arithmetic. We will take this up in future pedagogical discussions.