Even by One Minute

The Hook

In the middle of a book of Jewish law, the Rambam stops and teaches you how to subtract. Not metaphorically. He works an actual problem on the page: take two hundred degrees, fifty minutes and forty seconds away from one hundred degrees, twenty minutes and thirty seconds, and here is how you borrow, and here is what you are left with. It is the sort of thing you would expect in a manual for surveyors, and it is sitting inside the laws of sanctifying the new month, between rulings about witnesses and rulings about leap years.

And before he begins he does something even stranger. He warns us not to treat any of it as trivial, and he tells us in the same breath that the calendar we actually use today could be taught to schoolchildren in three or four days. He is about to spend chapters on a method nobody needs anymore, and he wants us to understand that this is not a detour. Why does a code of law slow down to teach arithmetic that has already been superseded?

Do Not Regard These Calculations Lightly

He opens by handing us a compliment. The court, he reminds us, made precise calculations and knew in advance whether the moon would be visible on a given night. And therefore, he says, we are assured that anyone with a proper spirit and heart, who desires words of wisdom and probes to grasp the mysteries, will want to know how those calculations were done. He is not commanding. He is describing the kind of person he is writing for, and by describing that person he invites you to become him.

Then comes the history, and it is unsparing. Among the sages of the nations who studied astronomy and mathematics in earlier eras there were many disagreements about these methods. Great wise men blundered. Things were hidden from them, doubts arose. There were those who calculated and calculated and never found the right approach; in the Rambam's own image, they plunged into mighty waters and came back holding a potsherd. That is a devastating sentence for a legal code, and it tells you how hard he considers the problem to be. Over the course of history several sages did find the way, and to this the Rambam adds that we also hold traditions received from our own sages, with proofs that were never committed to the texts in common circulation. So he has decided to lay out a working method, available to anyone whose heart moves him to take it up.

And then the warning. A person should not regard these calculations lightly merely because they are not needed in the present age. They are deep and abstract matters. They are the mystery of the intercalation, sod ha'ibbur, known to great sages who would not transmit it broadly but only to ordained and perceptive students. The fixed calendar we use now, in an age with no court to hear witnesses, is by contrast simple enough for schoolchildren. Read that carefully. He is not saying the hard thing is better than the easy thing. He is saying: do not let the ease of what you inherited convince you that the thing itself is small.

He goes further, in a passage that reads almost like a man defending his work to a critic who has not yet been born. Some gentile scholar or some Jewish sage trained in Greek wisdom will study these methods and notice an approximation here, a rounding there. Do not imagine we missed it. Wherever we were not exact, our own computations proved that the imprecision does not affect the determination of when the moon becomes visible, so it was not worth being exact about. And if a step seems inelegant, that too was deliberate: it buys an advantage elsewhere, it reaches a correct result without long computation, and it keeps a person who is not practiced in these matters from being flustered by work that would not change the answer.

Three Hundred and Sixty Degrees

Having said all that, he simply begins. The sphere of the heavens is divided into three hundred and sixty degrees and twelve constellations. Each constellation holds thirty degrees, starting from Aries. Every degree holds sixty minutes, every minute sixty seconds, every second sixty thirds, and you may keep dividing as far as you like. He names all twelve in order, ram and bull and twins and crab and lion and virgin and balance and scorpion and archer and goat and water carrier and fishes, and then he shows you how to read a number. A star at seventy degrees, thirty minutes and forty seconds is in Gemini, in the middle of its eleventh degree, because Aries used thirty and Taurus used thirty and ten and a half remain. A star at three hundred and twenty degrees is in Aquarius, in its twentieth degree.

Then the housekeeping rules that keep every later chapter honest. When you add, add like to like: seconds to seconds, minutes to minutes, degrees to degrees. Sixty seconds become a minute and go up. Sixty minutes become a degree and go up. Degrees are gathered in sets of three hundred and sixty, and whatever remains after you have taken out three hundred and sixty is the only figure that matters. The circle has no memory of how many times you went around it.

And then subtraction, and the sentence I would put on the wall. Whenever you wish to subtract one number from another, if the second number is greater than the first, even if it is greater by merely one minute, you must add three hundred and sixty degrees to the first number so that the subtraction can be performed. Even by one minute. The degrees can be identical, the whole quantity can look larger to the eye, and if a single minute has tipped the balance the other way you do not improvise and you do not round. You bring down the entire circle and start the operation properly. Then he walks the worked example through, converting a minute into sixty seconds, converting a degree into sixty minutes, and arriving at two hundred and fifty nine degrees, twenty nine minutes and fifty seconds.

Mean Motion and True Motion

The last part of the chapter states the physics, and it is the part that stays with you. The sun, the moon and the other five wandering stars each move at a uniform speed in its own orbit. They never incline to heaviness or to lightness. Whatever speed a body moved at yesterday is the speed it moves at today, and tomorrow, and every other day. Nothing up there is having a difficult week.

But their orbits, though they encircle the earth, are not centered on the earth. And so, the Rambam says, if you measure a body's progress against the sphere of the constellations, the sphere in which the earth is the center, its rate appears to change. On one day it seems to have covered more ground than on the day before, on another less. The steady speed is called its mean motion. The apparent speed, the one that runs fast and slow depending on where you are standing, is called its true motion, and it is the true motion that tells you where the moon really is in the sky tonight.

Then he fixes a starting point, because a calculation needs somewhere to begin: the eve of Thursday, the third of Nisan of the year he was writing, the seventeenth year of the two hundred and sixtieth cycle, the year four thousand nine hundred and thirty eight from creation, one thousand four hundred and eighty nine of contracts, one thousand one hundred and nine years after the destruction of the Second Temple. Every calculation in the chapters that follow counts forward from that Wednesday night. And finally he fixes a place. Since sighting the moon matters only in the Land of Israel, all these computations are centered on Jerusalem and the territory within six or seven days' travel of it, where the moon is regularly seen and the witnesses can actually reach the court. That region lies about thirty two degrees north of the equator, ranging from twenty nine to thirty five, and about twenty four degrees west of the center of the settled world, ranging from twenty one to twenty seven.

The Unifying Principle

Consider what the Rambam has done in one chapter. He has said that the heavens move at a perfectly constant rate, and that we cannot see that rate, because we are not standing at the center of the circle. What we see instead is an apparent motion that speeds up and slows down, and he tells us to build our whole reckoning on the apparent one, because that, and not the abstraction, is where the moon will actually be when a witness lifts his eyes.

The Baal Shem Tov's teaching that not a leaf turns over without a Providence intending it is usually heard as a claim about small things. Set it here and it becomes a claim about the difference between two descriptions of the same body. Nothing in the sky is irregular. The irregularity is in the relation between the mover and the place we happen to watch from. The Rambam is not correcting the heavens. He is correcting for us.

The Alter Rebbe describes a soul whose love for its Source is constant and uninterrupted, burning at a fixed rate whether or not the person feels it on a given Tuesday. What we experience is not that constant. What we experience runs hot and cold, and a person who judges himself by the felt rate concludes that he is inconsistent, unserious, going backward. The Rambam has given us the vocabulary. That is true motion, the apparent motion, measured from a place that is not the center. The mean motion never changed. And the Rebbe drew from this very section of the Mishneh Torah the point that the Rambam did not treat astronomy as foreign material grudgingly admitted into a holy book. He wrote it into the laws themselves, because a Torah that could not hold the actual mechanics of the sky would be describing some other world than the one we were given.

Modern Application

There is an anxiety in modern religious life about learning things that are not immediately useful. We want the practical shiur, the halacha we can act on before Shabbat. And here is the Rambam, in the middle of his life's work, teaching a method of computing lunar visibility for a court that has not sat in centuries, warning us explicitly that we will be tempted to skip it because we do not need it.

His answer is not that it might become useful again someday, although he believed it would. His answer is that these are deep matters, that they are the mystery of the calendar, and that the person he is writing for is the one who probes to grasp the mysteries. The value is not downstream of the application. Knowing how the thing actually works is itself part of keeping it.

And there is a smaller, sharper lesson in the arithmetic. Even if it is greater by merely one minute, add the three hundred and sixty. The whole quantity looked bigger. The degrees were on your side. A single minute in the smallest column reverses the operation, and if you eyeball it you will get an answer that is confidently, invisibly wrong. He is not being fussy. He is telling you where errors actually come from. Not from the parts you found difficult, which you checked twice, but from the parts that looked obvious.

The Closing

The Sfat Emet hears in chodesh the word chiddush, and reads the new month as the world being started again rather than merely continued. It is worth holding that beside a chapter which insists that nothing in the heavens ever varies. Both are true, and the tension between them is the chapter. The moon is doing exactly what it did last month, at exactly the speed it has always moved, and it is nevertheless new, because newness is not a property of the object. It is what happens when someone who is not at the center looks up and sees it.

The Rambam could have written down the rules and left the sky to the astronomers. Instead he stopped, taught us to count in sixties, taught us to carry the whole circle when a single minute demands it, and told us the difference between the speed a thing truly has and the speed it appears to have from where we are standing. Then he handed us a date and a city and said: begin here. That is a religious instruction as much as a mathematical one. Learn how it actually works. Do not round what you have not proven you may round. And when the movement of your own life looks uneven, remember that you are measuring from somewhere off center, and that the real rate has not changed at all.