Two Days Ahead of the Sun

The Hook

You are reading a code of Jewish law, and you have just come through chapters about witnesses and courts and the sanctification of the month, and now the Rambam hands you a number. Three hundred sixty five days, five hours, nine hundred ninety seven parts and forty eight moments. Then he tells you what a moment is: one seventy sixth of a part. Then he builds a season out of it, and an equinox, and a nineteen year cycle that closes without a crumb left over.

And at the very end of the chapter, after all of it, he turns around and says that both of the reckonings he has taught you are approximations, and that the real equinox happens about two days earlier than either one of them will tell you. He does not hide it in a footnote. He does not say it apologetically. He teaches you a beautiful machine with enormous care and then tells you where it is wrong, and expects you to keep using it. Why would a code of law do that?

A Second Opinion About the Sun

The chapter opens on a disagreement, and the disagreement is about eleven minutes. Among the Sages of Israel there is an opinion that the solar year is exactly three hundred sixty five and a quarter days, and there is a second opinion that it is slightly less than that. The second opinion is the one the Rambam sets out here in full: three hundred sixty five days, five hours, nine hundred ninety seven parts and forty eight moments, where a part is one of one thousand eighty divisions of an hour and a moment is one seventy sixth of a part.

The unit itself deserves a pause. A rega, a moment, is one seventy sixth of a chelek, and a chelek is one eighteenth of a minute. The Rambam is asking you to hold in your hands a division of time far finer than any instrument in his world could measure. Nobody had a clock that could see it. It exists only because the arithmetic requires it, and the arithmetic requires it because he wants the numbers to close.

And close they do. Working from this figure, the difference between a solar year and a lunar year comes out to ten days, twenty one hours, one hundred twenty one parts and forty eight moments. Run that difference through nineteen years, through the seven added months of the cycle, and the Rambam reports the result with visible satisfaction: there is no remainder at all. After every cycle a perfect correspondence is established between the solar years and the combination of ordinary and full lunar years. The sun and the moon, which have been drifting apart at eleven days a year since the beginning, come back into exact agreement and start again from nothing.

From there the seasons fall out mechanically. A quarter of that year is ninety one days, seven hours, five hundred nineteen parts and thirty one moments, and once you know when any one season begins you can walk forward season by season through the whole cycle by adding that figure again and again. It is the kind of calculation a person can actually do at a table with a pen, which is precisely why he gives it.

The First Spring, and Everything After

Then he sets the anchor. According to this reckoning, the vernal equinox of the first year of creation fell nine hours and six hundred forty two parts before the conjunction of Nisan. And, he adds, so does the vernal equinox of the first year of every nineteen year cycle since. The relationship that held at the beginning of the world holds again at the top of every cycle. The system does not merely repeat; it repeats the original.

The procedure that follows is short enough to memorize. Find how many complete years of the current cycle have passed. For each of them add ten days, twenty one hours, one hundred twenty one parts and forty eight moments. Gather the moments into parts, the parts into hours, the hours into days, exactly as you would when calculating a conjunction. Subtract the nine hours and six hundred forty two parts. Divide what remains by the length of a lunar month, take the leftover, add it to the conjunction of Nisan of the year you are asking about, and there is your equinox.

Notice what he has just handed an ordinary reader. Not a table. Not a ruling to be accepted on authority. A method, with its anchor, its increment, and its remainder, such that anyone patient enough with a pen can arrive at the answer himself and check the court. In a book of law this is an unusual kind of generosity.

What the Sages Relied On, and What He Only Thinks

Then comes the sentence that changes the temperature of the whole chapter. It appears to me, the Rambam writes, that the Sages relied on this calculation of the seasons for the institution of a leap year, back in the era when the High Court sat and would add a month because of the timing of the equinox or for other reasons. His reason is stated openly: this calculation is more accurate than the other one, and it bears a greater resemblance to what the astronomers have established than the opinion that the year is exactly three hundred sixty five and a quarter days.

Read that carefully. He is telling us which of two Talmudic opinions the highest court of Israel actually used to determine the calendar of the entire people, and he introduces it with nireh li, it appears to me. Not a tradition he received. Not a ruling he found. An inference, and he labels it as an inference, in the middle of the most authoritative code of law the Jewish people have ever produced. He is willing to reason his way to a conclusion about the practice of the Sanhedrin and equally willing to tell you exactly how much weight that conclusion can bear.

And then, in the last halachah, he takes something away. Both of these calculations, he says, are approximations. They are built on the mean rate of the sun's progress and not on its actual position in the sky. When you look at where the sun really is at these moments, the vernal equinox will have taken place approximately two days before the time either method produces. And he is careful to say that this applies to both opinions, the one he has just endorsed and the one he has just set aside.

So the more accurate reckoning is still two days off. The nineteen year cycle that closes without remainder closes without remainder around an average that the sun itself does not keep. He has given us a machine of extraordinary elegance and then told us, in the same voice, that the heavens do not run on it.

The Unifying Principle

The Alter Rebbe writes in Shaar HaYichud VehaEmunah that the world is not a thing that was created once and has been running since, but a thing being spoken into existence continuously, so that the divine word that made the heavens stands within them and renews them at every instant. If that is what is happening, then a mean solar year is a fiction of a specific and honorable kind. It is a number arrived at by averaging over a process that is not repeating itself but being made new each moment. No wonder the average misses by two days. The average is the best a human being can do with something that is not, in fact, a cycle.

The Baal Shem Tov pressed the same point from the other side with his teaching on particular providence, that a leaf turning over in the wind turns exactly as it does and for a reason, and that there is nothing in the world so small as to be handled by a general rule. Between those two ideas there is no room left for a mean sun. There is only the sun where it actually is, at this moment, which is the thing the Rambam says his calculation cannot give you.

And yet he gives the calculation. He gives it in detail, with its moments and its seventy sixths, and rules that the Sanhedrin most likely used it to decide when the year would be full. This is the real teaching of the chapter. A reckoning does not have to be true in order to be authoritative. It has to be honest, shared, checkable, and good enough for the purpose it serves. The court was not being asked to locate the sun. It was being asked to decide when the people would keep Pesach, and for that a slightly wrong number that everyone can compute is worth immeasurably more than a perfect one that no one can.

Modern Application

Almost everything a person uses to run a life is a mean value. The budget assumes a normal month, which never arrives. The schedule assumes a normal Tuesday. The understanding you have of your own marriage, of your own capacity, of how long a piece of work takes you, is an average built out of remembered instances, and it is off, and you know it is off. The temptation is to respond in one of two useless ways: to defend the average as though it were the truth, or to throw it out because it is not.

The Rambam models a third response, and it is the harder one. Build the instrument as well as you possibly can. Take it down to the seventy sixth of a part if that is what closing the arithmetic requires. Use it, rely on it, make real decisions with it. And say out loud, in the same breath, that it runs about two days ahead of the world. Both halves are required. Precision without the disclaimer becomes superstition. The disclaimer without the precision becomes an excuse.

The nireh li matters just as much. Here is a man writing a code meant to stand for a thousand years, who could simply have asserted which opinion the Sanhedrin followed, and who instead marks the place where his own reasoning begins so that anyone after him can see the seam and press on it. Most people, faced with a conclusion they have reasoned their way to and cannot prove, choose between overstating it and not saying it. He does neither. He says it, and labels it, and lets it stand.

The Closing

The Sfat Emet taught that the renewal of the month carries a light that the fixed order of nature cannot contain, which is why the new moon is a beginning and not merely a position. The whole apparatus of this chapter, the parts and the moments and the nineteen year cycle closing on itself with nothing left over, exists to serve a moment that is not really an arithmetical event at all. The numbers are how a court gets into the room. What happens in the room is something else.

The Rambam teaches you to compute the equinox to a seventy sixth of a part of an hour, and then tells you the sun is two days ahead of your answer, and expects you to go on using the calculation anyway. That is not a contradiction, and it is not humility as a decoration. It is a working description of every honest instrument a person has ever built. Get it as exact as you can, know precisely where it fails, and go set the festivals.