The Calculation That Decides Nothing

The Hook

The last chapter of Hilchot Kiddush HaChodesh opens with the Rambam explaining why he is about to write it. Our Sages said that among the questions put to the witnesses was, in which direction was the crescent of the moon inclined. Since the court asked that, he feels it appropriate to explain how the answer can be calculated.

Then, before the sentence has finished, he undercuts the whole enterprise. My statements will not be exact, he says, because this knowledge is of no consequence regarding the actual sighting of the moon. And in case you missed it, he says it again in the middle of the worked example seven halachot later. All the phases of this calculation are approximations and are not exact, because they are of no consequence regarding the sighting.

Consider where we are standing. Eighteen chapters of the most demanding arithmetic in the Mishneh Torah are behind us, mean motions carried to seconds of arc, tables of corrections, thresholds that decide whether a figure is added or subtracted. And the treatise ends with a chapter its author twice tells you does not decide anything. Something is going on here that is not astronomy.

The Tilt Under Everything

The chapter has to start further back than you expect. Before you can say which way a crescent leans, you have to know how the whole band of the constellations sits relative to the earth, and the Rambam lays it out plainly. The orbit of the sun does not run through the center of the earth from east to west. It is inclined above the equator, half of it toward the north and half toward the south.

It crosses the equator at exactly two points, the beginning of Aries and the beginning of Libra, opposite one another. So six constellations lean north, from the start of Aries to the end of Virgo, and six lean south, from the start of Libra to the end of Pisces. From Aries the incline grows until the beginning of Cancer, roughly twenty three and a half degrees north, then falls back to nothing at Libra, then grows southward to roughly twenty three and a half degrees at the beginning of Sagittarius, then returns. When the sun stands at the beginning of Aries or of Libra it inclines neither way. It rises due east, sets due west, and the day and the night are equal throughout the world.

Then the table, and the table is worth reading slowly because of what it does. Ten degrees along the band inclines four degrees from the equator. Twenty inclines eight. Thirty inclines eleven and a half. Forty inclines fifteen. Fifty inclines eighteen. Sixty inclines twenty. Seventy inclines twenty two. Eighty inclines twenty three. Ninety inclines twenty three and a half. Watch the increments. Four, four, three and a half, three and a half, three, two, two, one, a half. Motion along the band buys a great deal of latitude at the start and almost none near the turn. The last ten degrees of travel move the sun half a degree. That is the solstice, and it is why the ancients said the sun stands still there; it does not stop moving, it stops changing direction, and those are different things.

Numbers with units in them are handled by averaging between the two figures, so five degrees inclines two and twenty three degrees inclines nine. And the circle is folded rather than tabulated four times. Between ninety and a hundred eighty, subtract the number from a hundred eighty. Between a hundred eighty and two hundred seventy, subtract a hundred eighty from the number. Between two hundred seventy and three hundred sixty, subtract the number from three hundred sixty. What is left is a figure the table already knows.

Which Way the Crescent Points

Now the moon. Its distance from the equator is not one quantity but two, laid against each other. Take the inclination of the degree where the moon's true position falls, and note whether it is northerly or southerly. Then take the moon's first latitude, and note the same. If both lean the same way, add them. If they lean opposite ways, subtract the smaller from the larger, and what remains is the moon's distance from the equator, in the direction of whichever figure was larger.

The worked example is the same night the treatise has been using all along, the night of sighting on the second of Iyar. The moon's true position was already established at the nineteenth degree of Taurus, and that degree inclines about eighteen degrees north. The moon's own latitude is about four degrees south. Subtract the lesser from the greater and the moon stands fourteen degrees north of the equator, north because the eighteen was the larger of the two.

And only now does the chapter answer the question the court actually asked. If the moon is on the equator or within two or three degrees of it, it will appear due west and its crescent will point due east. If it is inclined north, it appears in the northwest and the crescent points southeast. If it is inclined south, it appears in the southwest and the crescent points northeast. And the farther the moon stands from the equator, the more pronounced the tilt.

Read those three rulings again and notice what the horns are doing. They never point where the moon is. A moon in the northwest carries a crescent aimed southeast, back across the sky toward the place it is not. The bright edge faces the sun, and the sun is below the horizon behind it, so the shape of the crescent is a statement about something you cannot see. Which is also, exactly, why the question is useful to a court. A man who invents a moon will put the horns somewhere plausible. A man who saw one will put them where the sun was.

The Rambam adds the second examination question in the same breath. They also asked how high the moon appeared, and this too can be worked out, from the arc of sighting: a short arc and the moon hangs low, a long arc and it rides high. Two questions, two calculations, and neither of them touches the length of the month.

Do Not Go Looking in Other Books

And then the treatise ends, and it ends with a declaration rather than a rule. We have now explained all the calculations needed for the sighting of the moon and for the examination of the witnesses, so that everything should be comprehensible to men of understanding and they should not lack awareness of any of the Torah's paths, and so that they will not venture out in search of it in other texts. Then a verse from Isaiah: seek out of the book of God, and read; none of these shall be lacking.

The force of that closing depends on remembering what the Rambam admitted earlier in these chapters, that the astronomical material came from the works of Greek scholars whose books are no longer in our hands. He took what was true wherever he found it, folded it into the Mishneh Torah, and then closed the door behind him. The point is not that the wisdom was ours to begin with. The point is that once it has been put to this use, holding a court to its work, it belongs inside the book of God and a student need not go elsewhere for it.

The Unifying Principle

Return to the strange fact we started with. Why compute, in the final chapter, a figure that changes nothing?

Because the question was never asked in order to fix the month. The court did not need to know which way the crescent leaned. The court needed to know whether the two men standing in front of it had actually been on that hillside at that hour. The direction of the horns is a test of a witness, not a measurement of a moon. And a judge who cannot compute the expected answer cannot hear the given one; he has no way to tell an honest man from a confident one.

Which explains the disclaimer, and turns it from an apology into a design specification. The model is approximate on purpose, because it is being held up against human eyes at dusk. If the Rambam had built it to the last second of arc, the court would have acquired a tool that convicts honest witnesses of small errors, and a fine enough instrument would eventually testify against every real person who ever looked at the sky. He builds it to the tolerance of the question. You cannot cross examine a human being with a device more precise than a human being.

The Talmud gives the principle its sharpest form when it reads the word atem, you, in the verse about the appointed times, and derives that the courts' declaration stands even when they have erred. The Lubavitcher Rebbe returned to that teaching constantly, and it is the hinge of this entire treatise. The heavens do not sanctify the month. A room full of men does. The astronomy exists to help them listen well, and the moment it is used to overrule them it has stopped doing its job.

Modern Application

We are the first people in history who can settle the crescent question instantly and to an accuracy the Rambam would have found comical. Which makes the temptation clearer, not weaker. The instrument is so good that we forget to ask what it was for, and we begin to use precision to replace judgment rather than to inform it.

Look at how often that goes wrong in ordinary life. A person brings you an account of something that happened to him, and you have the means to check the timestamps, the receipts, the distances. The Rambam would say run the calculation, absolutely, and then remember that its function is to help you hear him, not to score him. Approximately eighteen degrees. Approximately four. The chapter is teaching a judge how to hold an expectation loosely enough that a true report can differ from it and still be true.

And there is the matter of the horns. What a person reports about anything is oriented by where he was standing, and the tilt of his account is data about his position rather than error in the account. The Baal Shem Tov taught that a person is wholly in the place where his thoughts are. Read a witness that way and the direction he points becomes the most informative thing he says.

These chapters land on the fifth of Elul, which is the season for running the calculation on oneself. The trap of the month is precision: a reckoning taken to more decimal places than the question can carry, until a person is convicting himself of arcseconds. The Alter Rebbe pointed out that Israel counts by the moon because we resemble it, a body with no light of its own that wanes almost to nothing and returns. A ledger built to that tolerance is honest. One built finer than that is only cruelty with a table attached.

The Closing

The Sfat Emet reads the renewal of the moon as the model for every kind of beginning, that what is renewed is not replaced but revealed again, the same body turning its face back toward the light. The moon on the second of Iyar stood fourteen degrees north of the equator with its horns pointed southeast, at a sun nobody in that field could see. It had nothing of its own to show, and it was still the thing two men came down the hill to testify about.

And the court, having heard them, sanctified the month. Not the tables. The court.

The Rambam gives you the calculation and tells you twice it decides nothing, and then closes the book of times with a verse instructing you not to go looking anywhere else. Both halves are the same instruction. Know the expected answer as well as it can be known, hold it loosely enough that a living person can differ from it, and then let the person speak. That is what all nineteen chapters were building, and it turns out the last thing the heavens are for is teaching a room of human beings how to listen.