The Scroll of the Appointed Times

Words of the Master of the Calendar unto the Copying Scribe and unto the Sage of the Times

Hearken, O scribe who copies these words upon a new tablet: add not, nor take away; set not the earlier in place of the later; and exchange not one number for another because the two seem nigh. A drop copied amiss corrupts not its own place alone, but every drop that follows it, and every bowl that drinks from them.
And hearken thou also, O Sage of the Times who seeks to discover the name of a day: pass not over a labour because it is long, and say not, “A near answer is enough for me.” In this secret, nearness availeth nothing. He who is short by one unit at the beginning of the reckoning may find at its end another year, another month, and a cutlet of which his ear hath never heard.
The secret of the calendar lies open, and all its words are written before the eyes of every reader; yet it is a great secret. For not only that which is hidden is called a secret, but also that which lies open and whose paths few know how to follow to their end. For when the book is put into the hand of a reader, saying, What time is today, what time shall tomorrow be, and what wilt thou say tomorrow when a man asks thee, saying, What time was yesterday?—he shall answer thee and say, It is sealed and hidden; I know it not.
Every pair of days given to the calendar has this order: the first is the day of working and the second the day in question. The two days must not be exchanged. From every pair the work of the calendar shall bring forth one decree containing five things.

The First Tablet: How the Days Are Numbered

The Foundation Day

The Flying Spaghetti Monster chose one day and called it the Foundation Day.

According to tradition, the tablets were delivered in the days of Aššur-dān, king of Assyria¹, in the limmu year of Būr-Saggallē, governor of Guzana, in the month Simānu, on the day when the sun was eclipsed.
A note upon the note: for the binding reckoning of the calendar, the day of the solar eclipse just mentioned is the fifteenth day of June in the year 763 BCE, in the proleptic Julian calendar. That day is the seventh day of June in the same year according to the proleptic Gregorian calendar. In a continuous count in which the first day of January in the year 1 CE of the Gregorian calendar is day one, the day on which the tablets were delivered has the number −278,522. Therefore the number of the Foundation Day in that same continuous count is −15,055,671. All the dates that follow are computational extensions backwards in time. Where a calendar has no historical year bearing such a number, a year numbering that includes year zero is written; for the Julian calendar the rule that it has no year zero is preserved. For the Solar Hijri calendar the fixed cyclical reckoning is used; for the Chinese and Hindu calendars the algorithmic extension² prescribed for this book alone is used; and for the Maya Long Count the Goodman–Martínez–Thompson correlation is used. These are the dates of the Foundation Day:
Gregorian: 22 December 41,222 BCE.
Hebrew: 19 Sivan, year −37,460 Anno Mundi.
Hijri: 27 Rabiʿ al-Awwal, year −43,126 AH.³
Solar Hijri: 18 Azar, year −41,843 SH.⁴ ⁵
Julian: 28 October 41,221 BCE.
Traditional Chinese: cycle −643, year 57 of the cycle—Geng-Shen—the first month, day 22; the month is not intercalary.⁶
Hindu: year −41,162 of the Vikrama era, month Kārttika, lunar day 16; neither the month nor the day is intercalary.⁷
Indian Saka: 1 Pausha, year −41,299 Saka.
Thai Buddhist: 22 December, year −40,678 BE.
Ethiopian: 1 Hidar, year −41,227.
Coptic: 1 Hathor, year −41,503.
Japanese imperial: that day has no historical imperial era name; under a backward extension of the Kōki count: 22 December, year −40,561.
Minguo: 22 December, year −43,132 Minguo.
Baháʼí: year −43,064, day 11 of the month Masáʼil; by cycles: Kull-i-Shayʼ −119, Váḥid 14, year 9.⁸
Maya Long Count: −97.6.17.7.11.⁹
¹⁰ ¹¹

We do not say that there were no days before the Foundation Day, nor do we say that the numbering of days begins with it. The Foundation Day is a peg only, from which measurement is made in either direction.
The day on which the tablets were delivered is later than the Foundation Day by this count:

one hundred and forty-nine days;
and seven hundred and seventy-seven thousand days;
and fourteen million days.

The whole count is fourteen million, seven hundred and seventy-seven thousand, one hundred and forty-nine days.

The Foundation Day and the Days Before and After It

The Foundation Day shall receive the number one.
If a day is later than the Foundation Day:

count how many days later it is;
multiply the count by two;
add one.

If a day is earlier than the Foundation Day:

count how many days earlier it is;
multiply the count by two;
add nothing.

Thus the order of the days and their numbers shall be:

the third day before the Foundation Day: six;
the second day before it: four;
the first day before it: two;
the Foundation Day: one;
the first day after it: three;
the second day after it: five;
the third day after it: seven.

Thus every day, however early or however late, receives one number of its own, and no two days receive the same number.
The order of the days must not be inferred from the order of their numbers. The number serves to distinguish the days and feed them into the work of the Sauce; the order of the days themselves remains their order of earlier and later.

The Second Tablet: Names of the Numbers Given to the Work

When the Sage of the Times seeks the name of a day, two days stand before him:

the day on which he performs the reckoning;
the day whose name he seeks to know.

We do not wish to say in every line “the number of the day on which the reckoning is performed”. We shall therefore call it:
the Working Number.
Nor do we wish to repeat “the number of the day in question”. We shall therefore call it:
the Query Number.
The Sage shall also prepare three numbers:

The Distance Number

Count how many days lie between the day of working and the day in question.
If they are one and the same day, do not say that no number is found between them; write one.
If they are one day apart, write two.
If they are seven days apart, write eight.
This number shall be called the Distance Number.
The one is added so that no empty place remains in the reckoning.

The Sum Number

Add the Working Number to the Query Number.
Call the number obtained the Sum Number.

The Direction Number

If the day in question precedes the day of working, write one.
If both are the same day, write two.
If the day in question is later than the day of working, write three.
Call this number the Direction Number.
Let not the copyist confuse the Distance Number with the Direction Number. The distance tells how far apart the days are; the direction tells to which side one turns.

The Third Tablet: What Multiplication Is

When it is said, “multiply three by seven”, the meaning is:
Take the three seven times and add them:
three plus three plus three plus three plus three plus three plus three.
And so in every multiplication.
Yet the Sage of the Times need not add a great number to itself many thousands of times. He shall make a table of doublings:

write the number;
beneath it write twice that number;
beneath that write twice the double;
and so on.

Afterwards he shall add only the rows that must be added.
For example, to multiply a number by thirteen:

take eight times;
take four times;
take once;
add the three.

Eight, plus four, plus one, are thirteen.
Thus the Sage escapes a long labour without changing the meaning of multiplication.

The Fourth Tablet: What a Square Is

When it is said, “take the square of a number”, the meaning is:
Multiply the number by itself.
If the number is seven, its square is seven times seven.
To what may this be likened? Draw a square whose every side has the length of the said number, and divide it into small squares whose every side has length one. Count the small squares and see that the number is true and the craft of the reckoning correct.
When it is said, “take its square again”, the meaning is not to return to the first number that stood before thee, but to take the square already found and multiply it by itself.
Let not the Sage err in this. Many times we shall ask him to grind a number already ground, and not to return to the first grain.

The Fifth Tablet: The Great Number

We do not wish to keep in the reckoning numbers that grow without end. We shall therefore set a fixed bound and call it the Great Number.
Make it thus:

Write one.
Multiply it by two.
Multiply the result by two again.
After the two multiplications already made, continue and multiply the result by two another one hundred and twenty-five times. Thus thou hast multiplied by two one hundred and twenty-seven times altogether.
Fear not the number, though it is exceedingly great.
After these things, subtract one from the number thou hast found.

The number found is the Great Number, and so shall we call it.
There is no need to reckon it every time. It is enough that it be written once upon a tablet and well preserved.
The copying scribe shall count the doublings on two scrolls, lest he make one hundred and twenty-six or one hundred and twenty-eight. An error of one doubling changes the whole secret of the calendar.

The Sixth Tablet: What a Remainder Is

If a number is found greater than the Great Number, take the Great Number from it.
If it is still greater, take it away again.
Continue until a number remains that is not greater than the Great Number.
The number left shall be called the Preserved Remainder.
If the number was divided by the Great Number exactly and nothing remains, leave no empty place. Write the Great Number itself.
Thus the Preserved Remainder is always:

at least one;
and no more than the Great Number.

Whenever “preserve” is said hereafter, its meaning is: find the Preserved Remainder in this manner.
Writing the Great Number in place of a remainder means that nothing was left after all whole multiples had been removed.

The Ordinary Remainder

When it is said to take the ordinary remainder upon division by a number no smaller than one, the rule of the Preserved Remainder is not used.
The ordinary remainder is one of the numbers:
nothing, one, two, and so on to one less than the divisor.
The ordinary remainder is what is left after all whole multiples of the divisor have been taken from the number. If nothing remains, the remainder is nothing; if something remains, it is one or more, but always less than the divisor.
This rule shall be used when choosing a place among things set one after another, in opening the orders of the bowls, and in numbers written in places whose weights rise by powers of the Great Number.
Let the scribe not mix the two laws:

in the work of the Sauce itself—he shall not say there is no remainder; exact division is written as the Great Number;
in ordinary division—there are times when no remainder is left, and so it remains until another instruction adds one to it.

When a Greater Number Must Be Subtracted from a Smaller

At times one shall be told to subtract a number, but the number to be subtracted will be greater than the number from which it must be taken.
In such a case:

add the Great Number to the smaller number;
if it is still not great enough, add the Great Number once more;
continue as long as needful;
only then perform the subtraction;
preserve.

It must not be said that a number has fallen below nothing. In this secret one circles round the Great Number again, like a man who walks around a wall and returns to the gate from which he went out.

How to Find a Remainder Without Many Subtractions

If the number is very great, let not the Sage subtract the Great Number again and again until his spirit fails.
He shall do thus:

write the Great Number;
beneath it write twice the Great Number;
beneath that write twice the preceding number;
continue until the next double would be greater than the number from which one seeks to subtract;
subtract the greatest row that is not too great;
then try the row before it;
and so on to the first.

Thus in few acts he shall remove every whole multiple.
This is the selfsame reckoning. The way is shortened, but the number differs not.

The Seventh Tablet: The Five Stones

Every drop shall have five stones:

the Wheat Stone;
the Barley Stone;
the Salt Stone;
the Bitter Stone;
the Red Stone.

For the first drop give:

to the Wheat Stone: seventeen;
to the Barley Stone: twenty-nine;
to the Salt Stone: forty-three;
to the Bitter Stone: seventy-one;
to the Red Stone: one hundred and one.

To prepare the stones of the next drop, do thus:

The New Wheat Stone

Take the square of the old Wheat Stone.
Add three times the old Barley Stone.
Add the number of the new drop.
Preserve.

The New Barley Stone

Take the square of the old Barley Stone.
Add five times the old Salt Stone.
Add the old Wheat Stone.
Preserve.

The New Salt Stone

Take the square of the old Salt Stone.
Add seven times the old Bitter Stone.
Add the old Barley Stone.
Preserve.

The New Bitter Stone

Take the square of the old Bitter Stone.
Add eleven times the old Red Stone.
Add the old Salt Stone.
Preserve.

The New Red Stone

Take the square of the old Red Stone.
Add thirteen times the old Wheat Stone.
Add the old Bitter Stone.
Preserve.
Calculate all five new stones from the five old stones of that same preceding drop. Do not use a new stone already found to calculate another new stone. Only after all five have been found shall the five old stones be replaced together.
Thus make five stones for each of the forty-six drops.
The stones may be prepared once and kept on a separate tablet. They do not become a different number according to the days.

The Eighth Tablet: The Seven Hidden Drops

Before the first visible drop the Sage shall make seven drops that are not numbered among the drops of the Sauce.
They are called the Hidden Drops, because they are not poured into the bowls, yet the visible drops drink from them.
The one nearest to the first visible drop shall be called the First Hidden; the one before it the Second Hidden; and so on to the Seventh Hidden, which is farthest of all.
Set them in a row, one after another, thus:
the Seventh Hidden, the Sixth Hidden, the Fifth Hidden, the Fourth Hidden, the Third Hidden, the Second Hidden, the First Hidden, the first visible drop.
Do not reverse the order because the Seventh bears the greatest number. The number of a Hidden tells how far it lies from the first visible drop, not how near it is to it.
For every Hidden Drop add the Working Number once, and then add the numbers as I shall teach thee.

The First Hidden

three times the Query Number;
four times the Distance Number;
six times the Sum Number;
eight times the Direction Number.

The Second Hidden

five times the Query Number;
seven times the Distance Number;
ten times the Sum Number;
twelve times the Direction Number.

The Third Hidden

seven times the Query Number;
ten times the Distance Number;
fourteen times the Sum Number;
sixteen times the Direction Number.

The Fourth Hidden

nine times the Query Number;
thirteen times the Distance Number;
eighteen times the Sum Number;
twenty times the Direction Number.

The Fifth Hidden

eleven times the Query Number;
sixteen times the Distance Number;
twenty-two times the Sum Number;
twenty-four times the Direction Number.

The Sixth Hidden

thirteen times the Query Number;
nineteen times the Distance Number;
twenty-six times the Sum Number;
twenty-eight times the Direction Number.

The Seventh Hidden

fifteen times the Query Number;
twenty-two times the Distance Number;
thirty times the Sum Number;
thirty-two times the Direction Number.

To the First Hidden Drop add the five stones of the first visible drop; to the Second Hidden add the five stones of the second visible drop; and so on to the Seventh Hidden, to which the five stones of the seventh visible drop shall be added. After adding, preserve, and then grind each Hidden Drop seven times.
At every grinding:

take the square of the number;
add three times the number that stood before the squaring;
from the five stones belonging to the visible drop whose number is the same as that of the Hidden Drop, add the next stone in this order: Wheat, Barley, Salt, Bitter, Red, Wheat, Barley;
add the number of the grinding;
preserve.

Let not the scribe write merely “and so with the rest” without copying the order of the stones. Many have erred because they thought that after the Red Stone came a new stone, and did not return to Wheat.

The Ninth Tablet: Making the Forty-Six Drops

For brevity, we shall call the place of a drop in the order its Drop Number.
The first drop has number one, and the last number forty-six.
When a visible drop is made, the drop immediately before it, the third before it, and the seventh before it are required. Set the seven Hidden Drops before the first visible drop, and then count backwards without skipping.
Thus:

for the first drop: the previous drop is the First Hidden, the third before it is the Third Hidden, and the seventh before it is the Seventh Hidden;
for the second drop: the previous drop is the first visible drop, the third before it is the Second Hidden, and the seventh before it is the Sixth Hidden;
for the third drop: the previous drop is the second visible drop, the third before it is the First Hidden, and the seventh before it is the Fifth Hidden;
for the fourth drop: the previous drop is the third visible drop, the third before it is the first visible drop, and the seventh before it is the Fourth Hidden.

And so on. The scribe may draw a row of cells and write one drop in each cell.
For every drop first make the Drop Dough.
Add together:

the Wheat Stone multiplied by the Working Number;
the Barley Stone multiplied by the Query Number;
the Salt Stone multiplied by the Distance Number;
the Bitter Stone multiplied by the Sum Number;
the Red Stone multiplied by the Direction Number;
the previous drop;
three times the third drop before it;
five times the seventh drop before it;
the Drop Number.

Preserve.

Eleven Grindings

Grind the Drop Dough eleven times.
At each grinding do the following:

Take the square of the number obtained from the preceding grinding.
Add the preceding number multiplied by the grinding coefficient from the tablet below.
Add the preceding drop multiplied by the second coefficient.
Add the third drop before it multiplied by the third coefficient.
Add the seventh drop before it multiplied by the fourth coefficient.
Add one stone.
Preserve.

The grinding coefficients are:

at the first grinding: three, five, seven, eleven; and the Wheat Stone;
at the second: five, seven, eleven, thirteen; and the Barley Stone;
at the third: seven, eleven, thirteen, seventeen; and the Salt Stone;
at the fourth: eleven, thirteen, seventeen, nineteen; and the Bitter Stone;
at the fifth: thirteen, seventeen, nineteen, twenty-three; and the Red Stone;
at the sixth: seventeen, nineteen, twenty-three, twenty-nine; and the Wheat Stone;
at the seventh: nineteen, twenty-three, twenty-nine, thirty-one; and the Barley Stone;
at the eighth: twenty-three, twenty-nine, thirty-one, thirty-seven; and the Salt Stone;
at the ninth: twenty-nine, thirty-one, thirty-seven, forty-one; and the Bitter Stone;
at the tenth: thirty-one, thirty-seven, forty-one, forty-three; and the Red Stone;
at the eleventh: thirty-seven, forty-one, forty-three, forty-seven; and the Wheat Stone.

At every grinding do thus:

Take the number found at the preceding grinding and square it.
Add the number that stood before the squaring, multiplied by the first coefficient written in the row for that grinding.
Add the drop preceding the one being made, multiplied by the second coefficient written in the row.
Add the third drop before it, multiplied by the third coefficient.
Add the seventh drop before it, multiplied by the fourth coefficient.
Add the stone written in that row.
Add all these and preserve.

Let not the Sage exchange the four coefficients. The first multiplies the number being ground, the second the preceding drop, the third the third drop before it, and the fourth the seventh drop before it.
The number preserved after the eleventh grinding is the drop.
Finish one drop before beginning the next.

The Tenth Tablet: The Six Bowls

Set six bowls in a row.
So that we need not say again “the bowl standing first on the left”, give them names:

Bowl One;
Bowl Two;
Bowl Three;
Bowl Four;
Bowl Five;
Bowl Six.

The number of Bowl One is one, the number of Bowl Two is two, and so on to Bowl Six, whose number is six.
The names do not say that one bowl is more important than another.
For every bowl prepare an initial content:

add the Working Number;
add the Query Number multiplied by the Bowl Number;
add the Distance Number;
add the Sum Number;
add the Direction Number;
add the square of the following number:
- for the first bowl: seventeen;
- for the second: nineteen;
- for the third: twenty-three;
- for the fourth: twenty-nine;
- for the fifth: thirty-one;
- for the sixth: thirty-seven.
square the sum.
add the Bowl Number.
preserve.

The Eleventh Tablet: How the Drop Sets the Bowls in Their Order

Every drop sets the bowls in a different order.
The six bowls can stand in seven hundred and twenty orders.
Arrange all the orders thus:

First shall come all orders headed by Bowl One, then all orders headed by Bowl Two, and so on through those headed by Bowl Six.
Among orders beginning with the same bowl, put first the order whose second bowl has the smaller fixed name-number.
If the second bowl is also the same in both orders, compare the third bowl.
If it too is the same, compare the fourth, then the fifth and the sixth.
At the first place where two orders differ, the order containing the bowl with the smaller number shall come first.

Thus the first order shall be:
Bowl One, Bowl Two, Bowl Three, Bowl Four, Bowl Five, Bowl Six.
And the last order shall be:
Bowl Six, Bowl Five, Bowl Four, Bowl Three, Bowl Two, Bowl One.
To find the Order Number for a drop, do thus:

subtract one from the drop;
divide what remains by seven hundred and twenty;
take what remains after all whole multiples have been removed;
add one.

The number found shall be the Order Number.
Thus the Order Number is never less than one nor greater than seven hundred and twenty. If the drop itself is a multiple of seven hundred and twenty, the Order Number is seven hundred and twenty; we do not say that there is no remainder.
There is no need to write out all seven hundred and twenty orders. The Order Number may be opened by groups:

six groups of one hundred and twenty, to choose the first bowl;
five groups of twenty-four, to choose the second;
four groups of six, to choose the third;
three groups of two, to choose the fourth;
and at last, order the two bowls that remain.

The position of a bowl within the chosen order, from one to six, shall be called the Bowl Position Number. This is not the bowl’s fixed number.

The Twelfth Tablet: Pouring the Drop

Pour the drop into the first three bowls in the new order.

The First Pouring

Square the drop.
Add the Wheat Stone multiplied by the content of the first bowl.
Add three times the Drop Number.
Preserve.

The Second Pouring

Square the drop.
Add the Barley Stone multiplied by the content of the second bowl.
Add five times the Drop Number.
Preserve.

The Third Pouring

Square the drop.
Add the Salt Stone multiplied by the content of the third bowl.
Add seven times the Drop Number.
Preserve.
The last three bowls receive no separate pouring, but the drop shall enter them also during the stirring.

The Thirteenth Tablet: Stirring the Six Bowls

Before thou stirrest, copy the contents of all six bowls.
Let not the scribe cast away the old contents until the new contents of them all have been found.
The bowls stand in a circle according to the order fixed by the drop:

the bowl before the first in the order is the last in the order;
the bowl after the last in the order is the first in the order.

For every bowl, according to its place in the order fixed by the drop, do thus:

Take its old content.
Add twice the content of the bowl before it in the circle.
Add three times the content of the bowl after it in the circle.
Add the direct pouring, if it received one.
Add the whole drop.
Add one of the five stones according to the order of positions: Wheat, Barley, Salt, Bitter, Red, and Wheat once again.
Square the sum.
Add five times the product of the two neighbouring bowls, using their old contents.
Add the Drop Number multiplied by the Bowl Position Number in the chosen order.
Preserve.

After the six new contents have been found, replace all six together.

A Warning to the Two Who Perform the Work

The copying scribe shall not omit the word “together”.
The Sage of the Times shall not put a new content into the first bowl and then use it to calculate the second. All six bowls drink from the old contents, not the new.
Thus shall he do for all forty-six drops.

The Fourteenth Tablet: The Twelve Stirring Rounds After the Pouring

After the forty-sixth drop has been poured, let not the Sage say that the work of the bowls is finished.
He shall perform twelve further stirring rounds.
At each stirring:

Add together the contents of the six bowls.
Add one hundred and forty-nine times the Stirring Number.
Preserve. Call the number obtained the Stirring Order Number.
Subtract one from the Stirring Order Number just preserved.
Remove whole multiples of seven hundred and twenty from what remains until a number less than seven hundred and twenty remains. Add one to the remaining number.
The number found indicates the position of a bowl order among all the orders set one after another.
Take the bowl order standing at that position, the orders being arranged from earlier to later according to the rule written on the Eleventh Tablet.
Set the bowls in a circle according to the new order.
For each bowl add, using only the six old contents:
- its old content;
- three times the content of its previous neighbour in the circle;
- five times the content of its next neighbour in the circle;
- the preserved sum of the bowls;
- the Stirring Number;
- and the square of its Position Number in the new order.
Square the whole sum.
Add seven times the product of its two neighbours, using their old contents.
Preserve.
Only after all six new contents have been found, replace all six together.

For every bowl, according to its place in the order fixed for that stirring, do thus:

Take the bowl’s old content.
Add three times the old content of the bowl before it.
Add five times the old content of the bowl after it.
Add the sum of the six bowls preserved at the beginning of the stirring.
Add the Stirring Number.
Add the square of the bowl’s Position Number in the present order.
Add all these and square the sum.
Add seven times the product of the old contents of its two neighbours.
Preserve.

The bowls stand in a circle. Therefore the bowl before the first is the last, and the bowl after the last is the first.
All contents used in this reckoning are the old contents, which stood in the bowls before the stirring began. Let not the Sage use a new content of a bowl already calculated to find the content of another bowl.
Only after the new contents of all six bowls have been found shall he replace them all together.
After the twelfth stirring thou shalt have six results in thy hand.
These are the six bowls of the finished Sauce.

The Fifteenth Tablet: How the Bowls Are Asked

Give every question a seal.
The seal is no word of magic. It is a fixed number that prevents an answer given to one question from serving for another question also.
The seals shall be:

one: the way between Cutlet Gates;
ten: the beginning and end of Year Five Thousand;
eleven: the next year;
twelve: the preceding year;
twenty: the number of cutlets;
twenty-one: the division of the year into cutlets;
twenty-two: the names of the cutlets;
thirty: the number of months;
thirty-one: the lengths of the months;
thirty-two: the weaving of the months;
thirty-three: the names of the months.

There shall be no seal forty.

The Next Bowl

When a bowl is asked, “the next bowl” is the bowl following it in the circle of the order fixed by the forty-sixth drop.
For this purpose the order of the twelfth stirring is not used, nor is the fixed naming order One, Two, Three. If the bowl being asked is last in the order of the forty-sixth drop, the next bowl is the first in that same order.

The First Answer Number

When a bowl is asked a question:

Take the content of the bowl.
Add the seal.
Add one hundred and eighty-one.
Square the sum.
Add one hundred and seventy-nine times the content of the next bowl.
Add the seal.
Preserve.

This is the First Answer Number.

Fixing the Direction of the Answer Numbers

To determine whether the Answer Numbers shall rise or fall, do this once only:

Take the First Answer Number.
Add the seal.
Add one, because this is the number of the first additional number.
Add one hundred and ninety-three.
Square the sum.
Add one hundred and ninety-three times the First Answer Number.
Add one hundred and ninety-seven times the content of Bowl Six, according to its fixed name.
Preserve.

Call the number obtained the Direction Count.
If the Direction Count is odd, the answers run forward. If it is even, they run backward.

The Additional Numbers and Their Complete Circuit

The first additional number has number one, the second has number two, and so on.
If the answers run forward, the first additional number is the number immediately after the First Answer Number. The second additional number is the number immediately after that, and so on. After the Great Number, return to one.
If the answers run backward, the first additional number is the number immediately before the First Answer Number. The second additional number is the number immediately before that, and so on. Before one stands the Great Number.
Thus the row of Answer Numbers passes through every number, from one to the Great Number, exactly once before returning to its beginning. Moving one step forward or one step backward around a circle cannot return to the first place before passing through all the places.
It follows that every instruction which rejects an unfit Answer Number and asks for the next one shall necessarily reach a number fit to be accepted, provided that at least one such number exists.

The Sixteenth Tablet: Equal Choice Among Ways

¹²

Suppose admissible ways stand before the Sage, arranged one after another in their prescribed order. Call their number the Ways Count.
Do not at once take the ordinary remainder of the Answer Number divided by the Ways Count, for then the earlier ways may receive an excess share.

The Short Choice

If the Ways Count is not greater than the Great Number, do thus:

Find the greatest multiple of the Ways Count that is not greater than the Great Number. Call this multiple the Acceptance Bound.
Take the First Answer Number.
If it is greater than the Acceptance Bound, cast it away and take the next Answer Number in the complete circuit.
Continue until a number is found that is not greater than the Acceptance Bound.
Subtract one from it.
Divide what remains by the Ways Count, take the ordinary remainder, and add one.

The number found is the position of the chosen way among the ordered ways.
The Ways Count is at least one and no greater than the Great Number; hence the Acceptance Bound is at least one. The row of Answer Numbers passes through every number from one to the Great Number, and therefore it shall reach the Acceptance Bound or a smaller number before returning to its beginning.

The Wide Choice

If the Ways Count is greater than the Great Number, do thus:

Prepare one place. In this place one may write nothing, one, two, and so on to one less than the Great Number.
If the number of all possibilities in one place is smaller than the Ways Count, add a second place.
If two places are still not enough, add a third; and continue adding places until the number of all their combinations is not smaller than the Ways Count.
Call the number of places prepared the Places Count, and call the number of all possible combinations in them the Space Count.

Now fill the places:

Take the First Answer Number, followed by the additional numbers in their order, until thou hast as many numbers as there are places.
Subtract one from each number.
Write what remains from the first number in the first place, what remains from the second in the second place, and so on.
Each unit in the first place is worth one unit.
Each unit in the second place is worth the Great Number.
Each unit in the third place is worth the Great Number multiplied by itself.
Each unit in the fourth place is worth the preceding product multiplied once more by the Great Number; and so for every additional place.
Multiply what is written in every place by its weight, add all the results together, and add one.

Call the number obtained the First Wide Number. It is one of the numbers from one through the Space Count.
If the answers run forward, the next Wide Number is one greater; after the Space Count return to one. If the direction is backward, the next Wide Number is one smaller; before one stands the Space Count. Thus the row of Wide Numbers passes through every number in the Space Count once before returning to its beginning.
Find the greatest multiple of the Ways Count that is not greater than the Space Count. Call this multiple the Wide Acceptance Bound.
Test the First Wide Number, followed by the subsequent Wide Numbers in the fixed direction. Reject every number greater than the Wide Acceptance Bound. From the first number not greater than it subtract one, divide the remainder by the Ways Count, take the ordinary remainder, and add one.
This is the position of the chosen way among the ordered ways.
The Space Count is not smaller than the Ways Count, and therefore the Wide Acceptance Bound is at least one. The row of Wide Numbers passes through every number in the Space Count; therefore it must reach an acceptable number before returning to its beginning.
After rejecting a Wide Number, do not take a new group of Answer Numbers unrelated to the preceding one. All Wide Numbers are arranged in one circle, and each trial moves by one step only.

Where Each Choice Is Used

If the Ways Count is not greater than the Great Number, use the Short Choice.
If the Ways Count is greater than the Great Number, use the Wide Choice.
This rule applies to all admissible ways arranged one after another, and especially to:

year boundaries;
numbers of cutlets and months;
cutlet partitions;
sequences of month lengths;
month weavings;
and rows of names arranged one after another.

The choice of a bowl order does not use rejection of answers, but the direct rule of the seven hundred and twenty orders written on the Eleventh and Fourteenth Tablets.

The Seventeenth Tablet: The Cutlet Gates

The Foundation Day itself shall be a Cutlet Gate, and it is the middle gate.

The Gates After the Foundation Day

To find the distance from the middle gate to the first gate after it:

Take the Foundation Day as the day of working.
Take the first day after the Foundation Day as the day in question.
Make all forty-six drops and the six bowls.
Ask the first bowl with seal one.
Choose one number from nine hundred and twenty-two numbers arranged in increasing numerical order.
Add forty-one to it.

The distance shall therefore be between:

forty-two days;
and nine hundred and sixty-three days.

To find the distance from the first gate to the second, use the second day after the Foundation Day as the day in question. For the third gate use the third day, and so on. The day of working remains the Foundation Day.

The Gates Before the Foundation Day

To find the distance from the middle gate to the first gate before it:

Take the Foundation Day as the day of working.
Take the first day before the Foundation Day as the day in question.
Make the Sauce and ask with the same seal.
Choose in the same manner and add forty-one.

To find the distance from the first gate before the Foundation Day to the second gate before it, use the second day before the Foundation Day as the day in question, and so on.
Do not also use the first day after the Foundation Day for the first gate before it. The two sides receive different questions, and therefore their distances need not be equal.
This does not mean that the day used for the question is itself the gate. It serves only to draw the distance from the bowl.

Why the Gates Exist in Both Directions

Every distance is a number no smaller than one, and under this rule it lies between forty-two and nine hundred and sixty-three. Thus the later gates recede without bound, and the earlier gates recede backwards without bound. For every day, after a finite number of steps, there shall be found a gate no later than it and a gate no earlier than it.

The Eighteenth Tablet: Year Five Thousand

A year begins on the day after its opening gate and ends on its closing gate itself.
Thus a gate separating two years is the last day of the earlier year, and the day after it is the first day of the later year.
On the day of working, find all pairs of Cutlet Gates admissible as the beginning and end of a year, provided that:

the day of working is one of the days of the year—that is, later than the opening gate and no later than the closing gate;
at least six gate intervals lie between the two gates;
the length of the year is no less than two hundred and fifty-two days;
its length is no more than five thousand seven hundred and eighty-one days.¹³

Arrange the admissible years from shortest to longest. If two have equal length, put first the one whose opening gate is earlier.
Make the Sauce with the day of working also serving as the day in question.
Ask the first bowl with seal ten and choose one year by the rule of the Short Choice or the Wide Choice, according to the number of years standing for selection.
This year shall be called Year Five Thousand from the Creation of the World.

Why an Admissible Year Always Exists

Every interval between gates is at least forty-two days and no more than nine hundred and sixty-three days.
Six consecutive intervals are therefore at least two hundred and fifty-two days and no more than five thousand seven hundred and seventy-eight days long. Both bounds are permitted for a year, whose length is from two hundred and fifty-two to five thousand seven hundred and seventy-eight days.
Therefore one may take six consecutive intervals containing the day of working and obtain at least one admissible pair of gates. This follows from the bounds; it is not an assumption.

The Nineteenth Tablet: The Other Years

To find the year after a known year:

Take the closing gate of the known year as the opening gate of the next year.
Take this closing gate as the day in question; the day of working remains the day on which the reckoning is performed.
Find all later closing gates that form a year from two hundred and fifty-two through five thousand seven hundred and seventy-eight days long and containing six or more gate intervals.
Arrange the admissible years from shortest to longest.
Ask the first bowl with seal eleven and choose.

The next year begins on the day after its opening gate and ends on the chosen gate.
To find the year before a known year:

Take the opening gate of the known year as the closing gate of the preceding year.
Take this gate as the day in question.
Find all earlier opening gates that form an admissible year.
Arrange the admissible years from shortest to longest.
Ask with seal twelve and choose.

Let not the Sage of the Times leap from Year Five Thousand to a distant year by estimate. He shall go year by year, for every year is born from the gate of the one before it.
Every later year receives a number one greater than that of the year before it, and every earlier year receives a number one smaller than that of the year after it. After one year comes only one year, and no number is skipped. If the journey backward passes Year One, Year Zero follows it. The year before that shall be called One Before Zero, the one before that Two Before Zero, and so on. When these numbers are written in figures, a minus sign is placed before them.
Because every year is at least two hundred and fifty-two days long, reaching any day in question, however distant, requires a finite number of years. The years are contiguous and do not overlap: every day belongs to one year only.

The Twentieth Tablet: The Cutlets

Every year shall be divided into no fewer than six and no more than seventeen cutlets.
Every cutlet shall begin on the day after a gate and end at a gate. Every cutlet shall contain at least one gate interval, and therefore its length is at least forty-two days.
When the structure of a year is determined, the day of working is the day on which the reckoning is performed, but the day in question fed into the Sauce is the first day of that year. The same pair of days shall be used for all questions concerning the structure of that year.

The Number of Cutlets

Arrange all permitted numbers in increasing numerical order:
six, seven, eight, and so on through seventeen, after removing every number made impossible by the number of gate intervals in the year.
Ask the second bowl with seal twenty and choose.

Partition of the Gate Intervals

After the number of gate intervals in the year and the number of cutlets chosen for it are known, write every admissible partition as a row of numbers.
The first number shall be the number of gate intervals in the first cutlet.
The second number shall be the number of gate intervals in the second cutlet.
Continue thus through the final cutlet.
Every number shall be at least one, and the sum of all the numbers shall equal the number of gate intervals in the year.
If the day of working is a gate lying within the year, discard every partition that does not end a cutlet at that gate.
If the day of working is the year’s closing gate, no further deletion is required, because the year and its final cutlet already end there.
A day of working that is a gate belongs to the cutlet and year that end at it, not to those beginning after it.
After the invalid partitions have been removed, arrange the remaining partitions thus:

The partition whose first number is smaller comes earlier.
If the first numbers are equal, compare the second numbers.
If those too are equal, compare the third numbers.
Continue thus to the last place.
At the first place where two partitions differ, put first the partition containing the smaller number.

After all partitions have been arranged in this manner, ask the second bowl with seal twenty-one and choose the partition standing at the position found.

Names of the Cutlets

These are the seventeen names of the cutlets, in the order written:

Bronze
Fox
Kidney
Lagash
Thought
Four Parts of Nine
Palgurash
Papyrus Sedge
Cluster
Scorpion
Ash
Wheat
River
Laughter
Akkad
Horn
The Empty Jar

The name Four Parts of Nine is one name. The name Palgurash is merely a joining of sounds. The Empty Jar is one name and does not signify that the cutlet has no days.
Choose from the names written above as many distinct names as there are cutlets in the year, and arrange them in a row.
No name may appear twice in the same row, nor shall the same name be given to two cutlets in the same year.
Arrange all possible rows of names thus:

First compare the first name in each row according to its place among the names written above.
The row whose first position contains the name appearing earlier among the names above shall come earlier.
If the first names are equal, compare the second names.
If those too are equal, go to the third names.
Continue thus to the last place.
At the first place where two rows differ, put first the row containing the name that appears earlier among the names above.

Ask the fifth bowl with seal twenty-two and choose.

The Twenty-First Tablet: The Months

Every year shall be divided into no fewer than three and no more than forty-seven months.
Every month shall contain at least four days and no more than one hundred and twenty-three days.
The days of one month need not all stand together. They may enter among the days of other months, like one thread woven through a cloth of many threads.
When the structure of the year is determined, the day of working remains the day on which the reckoning is performed, and the day in question fed into the Sauce is the first day of the year.

The Number of Months

Divide the length of the year by one hundred and twenty-three. If days remain, add one month to the quotient. This is the smallest admissible number of months.
Divide the length of the year by four and discard the remainder. If the quotient is greater than forty-seven, write forty-seven. This is the greatest admissible number of months.
Arrange all numbers between the bounds, including the bounds themselves, in increasing numerical order.
Ask the third bowl with seal thirty and choose.

Lengths of the Months

Find all admissible ways to divide the days of the year among the months, so that every month receives no fewer than four days and no more than one hundred and twenty-three days.
Every way shall be written as a row of numbers:

the first number shall be the length of the first month;
the second number shall be the length of the second month;
and so on through the final month.

The sum of all numbers in the row shall equal the length of the year.
Arrange all admissible ways one after another thus:

Put first the way in which the first month is shorter.
If the lengths of the first months are equal, compare the lengths of the second months.
If those too are equal, compare the lengths of the third months.
Continue thus to the last month.
At the first place where two ways differ, put first the way containing the smaller number.

After all ways have been arranged in this manner, ask the third bowl with seal thirty-one and choose according to the rule of the Short Choice or the rule of the Wide Choice, as the Ways Count requires.
There is no need to write all the ways one by one.
The Sage of the Times may count how many ways begin with each permitted length of the first month, and subtract whole groups until he finds the group containing the chosen way.
Then he shall do the same with the length of the second month, after it the third, and so on through the final month.
But the shortcut must not change the position of the way among the ordered rows, nor select any way other than the one that would have been chosen had all the ways been written in full.

The Weaving of the Months

After the length of every month has been fixed, arrange the days of the year thus:

every month shall appear as many times as the number of days assigned to it;
the first appearance of the first month shall precede the first appearance of the second;
the first appearance of the second shall precede that of the third, and so on;
their final appearances shall likewise occur in that order.

Discard every weaving that does not satisfy both laws of appearance.
Write each weaving as a row of month numbers, one number for each day of the year.
A day bearing the name of the first month shall be written as one.
A day bearing the name of the second month shall be written as two.
Continue thus through the final month.
Arrange all admissible weavings one after another¹⁴ thus:

First compare the number written for the first day of the year.
The weaving containing the smaller number shall come earlier.
If the numbers are equal, go to the second day.
If they are equal there also, go to the third day.
Continue day by day to the end of the year.
On the first day where two weavings differ, put first the weaving containing the smaller month number.

For this ordering, the first month precedes the second, the second precedes the third, and so on through the final month.
Ask the fourth bowl with seal thirty-two and choose by the rule of the Short Choice or the Wide Choice.
The days are not chosen one by one. The choice is of the entire cloth.

Names of the Months

These are the forty-seven names of the months, in the order written:

Clay
Pomegranate
Elbow
Envy
Eridu
Toothpaste
Three Parts of Five
Karshumab
Tiger
Tin
Mist
Frankincense
Spindle
Rib
Carob
Uruk
Shame
Camel
Copper
Well
Yolk
Star
Honey
Spleen
Limestone
Joy
Fig
Nineveh
Frog
Pitch
Lamp
The Closed Door
Sesame
Nape
Silver
Susa
Storm
Donkey
Flour
Regret
Babylon
Tongue
Flax
Salt
Pear
Bow
Sand

The name Three Parts of Five is one name. Karshumab is merely a joining of sounds. Toothpaste and The Closed Door are each a single name.
Nothing about a month’s length, place, temper, cutlet, or year can be learned from its name. The month Clay is not made of clay, and the month Tiger does not devour the month that follows it.
Choose from the names written above as many distinct names as there are months in the year, and arrange them in a row.
No name may appear twice in the same row, nor shall the same name be given to two months in the same year.
Arrange all admissible rows of names one after another thus:

First compare the first name in each row according to its place among the names written above.
The row whose first position contains the name appearing earlier shall come earlier.
If the first names are equal, compare the second names.
If those too are equal, go to the third names.
Continue thus to the last place.
At the first place where two rows differ, put first the row containing the name that appears earlier among the names written above.

Ask the fifth bowl with seal thirty-three and choose by the rule of the Short Choice or the Wide Choice.
The first name chosen shall be given to the first month, and so on.

The Final Tablet: Knowing the Name of the Day

For every two days, in the order in which they were given:

the day of working;
the day in question;

first find Year Five Thousand on the day of working, and go from it year by year until the unique year containing the day in question is reached.
After the boundaries of the year have been found, calculate the structure of the year once, taking the day of working to be the first of the two days originally given, while the day in question fed into the work of the structure is the first day of that year.
After the structure has been found:

The year number is the number of the year reached by the Sage of the Times from Year Five Thousand.
The cutlet name is the name of the cutlet in which the day in question stands.
The day in the cutlet is the count of days from the beginning of the cutlet, its first day being counted as one.
The month name is the name of the thread appearing at the position of the day in question in the cloth of the months.
The day in the month is the number of times that same thread has appeared from the beginning of the year through the day in question, counting that day itself.

The bowls must not be asked what the day in the month is, what the day in the cutlet is, to which cutlet the day belongs, or to which month it belongs. These follow from the position of the day within the structure already fixed.
These are the five things that issue from the work of the day, and this is their order:

year number;
cutlet name;
day in the cutlet;
month name;
day in the month.

Let not the Sage add a sixth thing among these five. The person performing the reckoning may report separately additional things found along the way, but shall not insert them among the five things that constitute the result for the day.

A Statement of Existence and Uniqueness

For every two days, in the order in which they were given, one result is obtained, because:

every day has one unique Day Number;
every operation of the Sauce, every drop, bowl, and stirring is determined only by the numbers and contents found before it;
all things standing for selection are arranged in an explicit order;
every choice involving rejection of answers terminates, because the row of Answer Numbers passes through every number before returning to its beginning, and the row of Wide Numbers passes through every number of the space before returning to its beginning;
every interval between gates is no shorter than forty-two days and no longer than nine hundred and sixty-three days, and therefore any day can be reached after a finite number of gates;
the years are contiguous and do not overlap, and therefore the day in question lies in exactly one year;
and the structure of the year is calculated from two fixed days—the day of working and the first day of the year—so two days in the same year under the same day of working do not give that year two different structures.

This is the law of all days past and future, and it has neither first nor final bound.

The Seal of the Master of the Calendar

The copying scribe shall guard the words.
The Sage of the Times shall guard the deeds.
The copyist who errs changes the law for all who come after him.
The Sage who errs corrupts the one day about which he asked, and at times many days after it if he kept his results.
Let no man say: “I am accustomed to the calendar, and therefore I shall know the morrow without reckoning.”
For month is woven through month, the cutlet is cut at the gate, year is born from year, the drop drinks from seven drops before it, and every bowl drinks from its five sisters.
Forty-seven names of months have I found, and seventeen names of cutlets have I found; I neither diminished them, nor doubled them, nor reversed their order.
Every way have I ordered, every neighbourhood have I compassed in a circle, and every circuit of answers have I closed only after it had passed through all its places.
This is an open secret, and great is its secret, for the sight of man falleth short of its labour.

External Links

The Hebrew original — מגילת העיתים
Computational implementation of the calendar on GitHub — software that implements the calendar algorithm and calculates dates according to the rules set out in this scroll.

Permission to Copy

This document may be copied and distributed in its entirety and without alteration.

Aššur-dān III is meant.↩︎
The term “algorithmic extension” does not designate a single generally accepted method. To reproduce the date numerically, the algorithm and the version used for the calculation must be regarded as part of the definition of the date; the result does not imply that every proleptic extension of the Chinese or Hindu calendar will yield it.↩︎
The Hijri conversion here uses the tabular Islamic calendar, extended algebraically backwards; it does not mean that the beginning of the month is determined by observation of the moon.↩︎
This date is a proleptic extension of the Solar Hijri calendar according to the astronomical model used in the calculation; there is no agreed historical date for so remote a period, and other methods of calculation may produce a different result.↩︎
Clarification: notwithstanding the neighbouring note’s reference to an “astronomical model”, the reckoning actually used here is the 2,820-year cyclical Persian arithmetic calendar. The result 18 Azar −41,843 is the result of that extension; it is not obtained by determining Nowruz from the instant of the vernal equinox.↩︎
This is a proleptic date calculated by extending the rules of the astronomical Chinese calendar backwards. The calendar did not exist in this period, and the resulting month and day depend on the astronomical model and on how the extension is defined.↩︎
This is a proleptic date according to the Hindu computational method used here. Several systems and sets of rules exist for the Hindu calendar, so at such a remote extension this date has no standing as a single agreed conversion.↩︎
The Baháʼí date is an arithmetic extension backwards of the Badíʿ count, not a historical application of the rules of the calendar to this period. The decomposition of years into Kull-i-Shayʼ and Váḥid has likewise been extended arithmetically here to negative counts.↩︎
Euclidean division is used for the negative Long Count notation: the negative sign belongs to the baktun component, while the components following it remain non-negative remainders within their respective ranges. Thus −97.6.17.7.11 is not the negation of the positive number 97.6.17.7.11. The GMT correlation used is 584283.↩︎
The correspondences between calendars designate the same day in the continuous day count, not equality between the instants at which days begin and end in the different calendars. In particular, calendars whose day begins at sunset or under some other local rule do not divide time at the same boundaries as the Gregorian day.↩︎
Negative year numbers are a computational extension only. When a date is written in the form “BCE”, the historical numbering without a year zero is retained; in calculations where the year is written as a signed integer, year zero lies between year 1 and year −1. Thus astronomical year −41221 must be distinguished from 41,222 BCE.↩︎
The title “equal choice” is not true in its probabilistic sense for the Wide Choice. The digits of the First Wide Number are taken from consecutive Answer Numbers and therefore are not independent choices in base Great Number. Consequently, the First Wide Number cannot take all values in the Space Count with equal probability; when the number of ways is sufficiently large, there are even ways that cannot be chosen at all. This does not impair determinism or termination of the algorithm, but it does invalidate the claim that the choice among all admissible ways is equal.↩︎
Only in this place is 5,781 written; the binding upper bound is 5,778 days, as fixed below and as follows from six gate intervals each at most 963 days long: 6×963=5,778.↩︎
Computational note: the instruction to write down all admissible weavings and arrange them defines the selection, but it cannot be carried out by actually generating the list. Even in the smallest case permitted by the rules of the calendar—a 252-day year divided into three months of 123, 123, and 6 days—there is already a subset containing more than 5.3×10^79 admissible weavings. This number is enormous in the extreme: even examining one weaving per Planck time would take more than 10^18 ages of the universe.
Instead, the chosen weaving can be found directly from its position in the order. Construct it from left to right; at each position, test the legal month numbers in increasing order and use dynamic programming to calculate how many admissible completions exist for each possible prefix. If the desired position is greater than the number of completions for a given prefix, subtract that number and skip the entire group; otherwise fix that month number and continue to the next position. The computational state is determined by the number of occurrences remaining for each month and by which months have already opened and which have already ended, so the completions can be counted without generating them.
This shortcut is equivalent to explicit enumeration because, in lexicographic order, all weavings with a given prefix form one contiguous block. Skipping a block by its number of members therefore preserves the exact position of the desired weaving; by induction over the positions in the row, the result is the same weaving that would have been selected from the complete list.↩︎