Know That You Have Erred

The Hook

Chapter nine opens with a confession. There is a disagreement among the Sages of Israel about the length of a solar year. Some hold it is three hundred and sixty five days and a quarter, which is to say six hours. Others hold it is a little less than that. And then, before anyone can find this reassuring, the Rambam adds that the wise men of Greece and the wise men of Persia disagree about it as well. Nobody has the number.

Now watch what he does with that. He does not soften it, he does not adjudicate it, and he does not tell you that the exact figure hardly matters because Rosh Chodesh is set by the court anyway. He takes the disputed number, adopts one of the two positions, and proceeds to build an apparatus of terrifying precision on top of it. Ninety one days and seven and a half hours from one season to the next. One hour and four hundred and eighty five parts of remainder after every nineteen year cycle. A lunar month of twenty nine days, twelve hours, and seven hundred ninety three parts. Add, group the parts into hours and the hours into days, subtract seven days, nine hours, and six hundred and forty two parts, and you will know the hour and the day of the vernal equinox for any year you like, forever.

A person raised on a certain kind of religious rhetoric expects a chapter like this to end by saying that the sun and moon are only shadows of a deeper truth. It does not. It ends by telling you that if your answer comes out too large, you have made a mistake and should go back and count again.

A Year Nobody Agreed On

Begin with the arithmetic, because the arithmetic is the argument. On the view that the solar year is exactly three hundred sixty five and a quarter days, a nineteen year cycle leaves a remainder of one hour and four hundred eighty five parts over the lunar reckoning. That small surplus is the whole reason the seasons drift against the months, and it is the number you carry forward every time you compute.

Between the start of one season and the start of the next there are ninety one days and seven and a half hours, exactly a quarter of the year, and here the Rambam offers something that reads like a promise. When you know the date and hour of the beginning of one season, you can calculate the next by adding that amount. And the next. And so on forever. One fixed point, honestly obtained, generates the entire future.

Then he anchors the seasons in the sky rather than in the calendar. The equinox of Nisan is the hour and the unit at which the sun enters the beginning of Aries. The solstice of Tammuz is when it stands at the beginning of Cancer, the equinox of Tishrei at the beginning of Libra, the solstice of Tevet at the beginning of Capricorn. These are not four festivals and they carry no obligations. They are four positions of a body in space, and the Rambam wants them located to the part.

And then, quietly, he gives the number that ties the whole system to the first week of the world. In the first year of creation, the vernal equinox fell seven days, nine hours, and six hundred forty two parts before the conjunction of Nisan. Seven, nine, six hundred forty two. Every calculation in the chapter runs back through that constant. He is not telling a story about creation here. He is handing you a fixed reference point and telling you that from that point the sun has never once been off schedule.

Four Hours the Sun Is Allowed

The procedures come next, and they are laid out the way a craftsman lays out a jig. Count the completed nineteen year cycles. Add an hour and four hundred eighty five parts for each. Group parts into hours, hours into days. Subtract the constant. Add the remainder to the conjunction of Nisan in the first year of that cycle, and you have the equinox. For a particular year inside the cycle, add ten days, twenty one hours, and two hundred four parts for each completed year, subtract the constant, divide out whole lunar months, and add what is left to that year's conjunction.

And then, in the middle of a paragraph of instructions, comes the sentence that should stop a reader. According to this calculation, the vernal equinox will always fall either at nightfall, at midnight, at daybreak, or at noon. The summer solstice will always fall at half past seven or half past one, morning or night. The autumn equinox will always fall at nine or at three, day or night. The winter solstice will always fall at half past ten or half past four.

Read that slowly. The sun, in this reckoning, is permitted four hours per season and no others. Four for Nisan, four for Tammuz, four for Tishrei, four for Tevet. If you compute the summer solstice and get nine in the morning, you have not discovered an unusual year. You have discovered that you cannot add. The system carries its own proof inside itself, and it is the kind of constraint that only emerges from a calculation clean enough to have consequences.

For the weekday there is a second cycle, of twenty eight years rather than nineteen. Count the complete years from creation, divide them into groups of twenty eight, add a day and six hours for each year remaining, then add three days, divide into weeks, and count the remainder from nightfall on the first day. Why the three days? Because the first equinox of the year of creation took place at the beginning of the fourth day, the day the luminaries were hung. Everything in the chapter is measured from the moment the clock was started.

Then he works an example, and the example is the best evidence that he means all of this literally. Take the year four thousand nine hundred thirty. Divide by twenty eight; the remainder is one, which gives one day and six hours. Add three days. The vernal equinox falls on the night of the fifth day at midnight. Add seven and a half hours and the summer solstice falls on Thursday, an hour and a half after daybreak. Add again and the autumn equinox is Friday, nine hours after daybreak. Add again and the winter solstice is the night of the sixth day, four and a half hours after nightfall. Every one of those four times is one of the permitted hours. He is showing his work and letting you check him.

Counting to the Date, and the Check at the End

The last movement asks a plainer question: not which hour, but which day of which month. Find the weekday of the equinox. Find the weekday of Rosh Chodesh Nisan. Count how many years of the nineteen year cycle are complete, add eleven days for each, add a further seven in the present era, divide the sum by thirty, and count the remainder forward from Rosh Chodesh Nisan. If you land on the weekday of the equinox, you are done. If not, add a day, or two, or three, until you reach it. And if the year is a leap year, do not begin from Rosh Chodesh Nisan at all. Begin from Rosh Chodesh of the second Adar.

He works this one too. Four thousand nine hundred thirty is the ninth year of the two hundred sixtieth cycle, so eight complete years, eight times eleven is eighty eight, plus seven is ninety five, divided by thirty leaves five. Rosh Chodesh Nisan is Thursday; five days on is Monday. But the equinox is Thursday, so keep adding until Thursday, and the answer is the eighth of Nisan.

And then the final halachah, and it is unlike anything else in the treatise. Although we said to keep adding days until you reach the weekday of the equinox, the Rambam writes, you should never have to add more than one, two, or three, or in a most unusual case four. If you find that you need more than that, know that you have erred in your reckoning, and go back and calculate carefully.

He has built a machine and then handed you a way to know it has broken. Not a rebuke, not a threat, not a warning that a mistaken calendar will cost you the festivals. Just a quiet range, and the instruction that stepping outside it means the error is yours and the remedy is to count again.

The Unifying Principle

Here is what the chapter is actually claiming, and it is a large claim. The world is exact, and human beings are not, and the way you live inside that gap is by carrying instruments that tell you when you have slipped.

Notice the shape. The chapter begins with a disagreement it cannot resolve, proceeds with a precision it fully intends, and ends with an error check. That is not a contradiction. That is a description of every serious relationship with truth that a person will ever have. You will not have the final number. You will still be required to work to the part. And you will be given some way of noticing when your work has come apart, which is not the same as being given certainty.

The Alter Rebbe writes in Shaar HaYichud VeHaEmunah that the world is not a thing that was made once and left running, but a thing being spoken into existence continuously, letter by letter, and that if the divine speech ceased for an instant nothing would remain of the heavens or the earth. Read chapter nine with that in your ear and the arithmetic changes character. Ninety one days and seven and a half hours is not the description of a mechanism grinding on by itself. It is the regularity of something being said again, deliberately, without variation, and the reason the sun has only four hours available to it is that it is not improvising.

The Baal Shem Tov taught that a leaf turning over in a stream is turning because it was decided, and that nothing anywhere is loose. His students found that teaching hardest to hold exactly where the world looks most mechanical, and the four permitted hours are as mechanical as anything in the Torah. That is the point. The regularity is not the absence of attention. It is what sustained attention looks like from underneath.

And the Lubavitcher Rebbe often noted how much of himself the Rambam invests in these chapters, this long stretch of pure astronomy set down as Torah in the same voice as the laws of Shabbat, and later in this very treatise conceding openly that these matters are the wisdom of the seasons and of geometry, and that one may go and learn them from the books of the wise men of the nations. He does not fence off the calculation as secular and the sanctification as holy. The reckoning is the mitzvah. Knowing the sky accurately is a religious act.

Modern Application

Two things follow, and I find the second one harder every year.

The first is that unresolved disagreement is not an excuse to stop working. The chapter's opening line would function beautifully as a shrug. The Sages disagreed, the Greeks disagreed, the Persians disagreed, so who can say anything. Instead the Rambam picks up a position, states which one he is using, and computes to the part. We live surrounded by genuine uncertainty and we have learned to use it as an anaesthetic: nobody really knows, the experts disagree, so nothing can be asked of me. This chapter refuses the move. Adopt a position honestly, name it, and then be exact inside it. The disagreement in the first halachah does not license a single sloppy calculation in the seven that follow.

The second is the error check, and it is a spiritual instrument disguised as a footnote. What the Rambam gives at the end of chapter nine is a way of catching a mistake without first having to detect the mistake itself. You will not notice that you mislaid a remainder six steps back. You will notice that you are adding a sixth day. Most people have no such instrument in the parts of life that matter most. We know when a marriage has failed, when the resentment has hardened, when the work has gone hollow, and we know it far too late, because we had no small early number to watch. Set one. If you find yourself explaining the same thing to the same person for the third time. If you cannot remember the last conversation with your child that was not logistics. If a week has passed without one honest hour of learning. That is your fourth day. Do not argue with it. Know that you have erred, and go back and count carefully.

The Closing

The Sfat Emet taught that renewal is not something that happens to time from outside but the condition on which time is granted at all, that each hour is issued fresh and can therefore be entered fresh. The four hours the sun is allowed each season are, in that light, not a cage. They are a promise that the thing is being held steady on purpose, and that a person who computes carefully will find it where it was said to be.

A chapter that opens by admitting nobody knows the length of a year, and closes by telling a student that if his answer runs past four days he has made a mistake and should count again, is not a chapter about astronomy. It is a chapter about how to be a serious person in a world you cannot fully measure. Take a position honestly. Work to the part. Keep one small number where you can see it, so that the day the arithmetic goes wrong you will know before the season has passed. The sun has four hours and it keeps them. The least we can do is check our work.