The Body That Is Not There

The Hook

Fifteen chapters of arithmetic have all been in the service of one very physical event. A man stands outside Jerusalem at dusk on the thirtieth of the month, sees a thin curve of light, and comes to court to say so. Everything the Rambam has built, the means and the elongations and the tables of angles, exists to let the court know in advance whether that testimony is plausible.

And now, this far into it, he introduces two things that cannot be seen at all. The moon's orbit is tilted against the sun's, and any two tilted circles cross at exactly two points, opposite one another. He calls the first the head and the second the tail. Nothing is at either one. They are geometry, the bare fact of an intersection, and they have no light and no body and no location a witness could ever be walked out to and shown.

The Rambam gives them a daily rate of three minutes and eleven seconds. He gives their travel over ten days, a hundred days, a thousand, ten thousand. He gives their position on the Thursday night that anchors his tables. He treats an intersection as though it were an object with an itinerary, and by the end of the chapter you understand that without it you cannot say where the moon is in the only dimension that decides whether it will be seen.

The Head and the Tail

The definition comes first and it is careful. When the moon stands at one of the two crossing points, it is revolving in the same plane as the sun. As it departs from a crossing point it begins to run either north or south of that plane. The point at which it begins to incline northward is the head. The point at which it begins to incline southward is the tail. Neither name describes a thing. Each names a moment in a cycle where a relationship changes sign.

Then a detail that is easy to read past and impossible to recover from if you do. The head revolves at a uniform pace in opposition to the movement of the sphere of the constellations. It goes from Pisces toward Aquarius, backward through the order everything else runs forward in, and it does this continuously and without increase or decrease. Every other figure in these chapters advances. This one retreats, and it retreats at a rate so steady that the Rambam can hand you a table and be done with it.

The table is the usual kind. Three minutes and eleven seconds in a day. Thirty one minutes and forty seven seconds in ten days. Five degrees, seventeen minutes and forty three seconds in a hundred. Fifty two degrees, fifty seven minutes and ten seconds in a thousand. One degree, thirty two minutes and nine seconds in twenty nine days, which is the interval he keeps returning to. And the anchor, the mean of the head on the Thursday night that begins his reckoning, is one hundred and eighty degrees, fifty seven minutes and twenty eight seconds.

Then the instruction that carries the retrograde motion into the arithmetic. Calculate the mean progress of the head the same way you calculated the mean of the sun and the mean of the moon, and then subtract that mean from three hundred and sixty degrees. What remains is the location of the head. The forward number you just built is not the answer. It is the thing the answer is measured against, and the subtraction is where the backward motion of the head actually shows up on the page. A man who computes the mean flawlessly and then writes it down as the position has done every step right and produced a figure that is running the wrong way around the sky.

The Rambam works it once. Twenty nine days after the starting point, the mean of the head is one hundred and eighty two degrees, twenty nine minutes and thirty seven seconds. Subtract from three hundred and sixty and one hundred and seventy seven degrees, thirty minutes and twenty three seconds remain. The seconds are of no consequence. The head sits at twenty seven degrees and thirty minutes of Virgo, and the tail sits directly opposite at twenty seven degrees and thirty minutes of Pisces. There is always an even half of the sphere between them, which means the tail is always seven constellations along from the head at exactly the same degrees and minutes.

Five Degrees, and Never More

With the head established, the Rambam turns to the question the whole chapter exists for. The moon's inclination north or south of the sun's plane is called its latitude. If the moon stands at the head or at the tail, in degrees and in minutes, it has no latitude at all. If it has passed the head and is proceeding toward the tail, its latitude is northerly. If it stands before the tail and is proceeding toward the head, its latitude is southerly.

And then the boundary. The latitude will never exceed five degrees, north or south. The pattern is fixed. From the head the moon diverges a little, and the divergence keeps growing until it reaches five degrees, and then it begins closing again until there is no latitude at all when it arrives at the tail. Then it diverges to the south, out to five degrees, and returns. The moon is always leaving the plane and always coming back, and it never gets more than five degrees away from it in the entire history of its wandering.

To put a number on it you take the true position of the moon, which the previous chapter taught you to find, and subtract the position of the head. What remains is called the course of the latitude. Between one degree and a hundred and eighty, the latitude is northerly. Above a hundred and eighty, southerly. At an even hundred and eighty or an even three hundred and sixty, there is no latitude at all, because the moon is standing at one of the crossings.

Then the table of angles, running only to ninety. Ten degrees of course gives fifty two minutes. Twenty gives one degree and forty three minutes. Thirty gives two degrees and thirty minutes. Forty gives three degrees and thirteen minutes, fifty gives three degrees and fifty minutes, sixty gives four degrees and twenty minutes, seventy gives four degrees and forty two minutes, eighty gives four degrees and fifty five minutes, and ninety gives an even five degrees. The increments shrink as the number climbs, which is what a curve flattening toward its maximum looks like written out as a list. For a course with units as well as tens, take the average increase per degree and add the proportion, so that a course of fifty three, sitting three degrees above fifty, gives three degrees and fifty nine minutes.

The table stops at ninety because it does not need to go further. A course between ninety and a hundred and eighty is subtracted from a hundred and eighty. A course between a hundred and eighty and two hundred and seventy has a hundred and eighty subtracted from it. A course between two hundred and seventy and three hundred and sixty is subtracted from three hundred and sixty. The Rambam works all three: a course of a hundred and fifty gives the angle of thirty, a course of two hundred gives the angle of twenty, a course of three hundred gives the angle of sixty. One quarter of a circle carries the whole of it, because the shape repeats four times and he saw no reason to print it four times.

Friday Night, Toward the South

The chapter closes where the last one closed, on the night of the second of Iyar. The true position of the moon that night was established already at eighteen degrees and thirty six minutes of Taurus. The position of the head has now been established at twenty seven degrees and thirty minutes of Virgo. Subtract the head from the moon and the course of the latitude is two hundred and thirty one degrees and six minutes.

The minutes are of no consequence, as always. Two hundred and thirty one falls in the third band, so a hundred and eighty comes off it and fifty one remains. Fifty one is one degree above fifty, and the increase in that stretch is three minutes per degree, so the angle is three degrees and fifty three minutes. That is the latitude of the moon at the beginning of that night. It is southerly, because the course was larger than a hundred and eighty.

Notice what the answer is not. It is not a place. It is not where the moon is. It is how far the moon has drifted off the plane the sun keeps, and in which direction, and the whole apparatus of the chapter exists to produce that one small number. Three degrees and fifty three minutes to the south. On a night when a court is waiting to hear whether two men saw something, that quantity is part of what decides whether they could have.

The Unifying Principle

The chapter is built on two facts that sit oddly together. The first is that the reference point is invisible. The head is not a body and never was. The second is that the deviation is bounded. Five degrees, and no more, ever, in either direction. The moon is permitted to wander, and the wandering is real and has to be measured, and it is also fenced.

The Alter Rebbe writes in the Tanya that a sin does not sever the soul from its source, and that the separation a person feels is an obscuring rather than a cutting. The connection is not broken. It is covered over, and the covering has a limit set by something other than the person's own behavior. Read the ninth halachah of this chapter as a picture of that and the arithmetic acquires a strange warmth. The moon leaves the plane. It leaves it every single month, twice, once to each side. And the leaving has a ceiling that the moon did not choose and cannot exceed, and the return is not an achievement but a feature of the orbit.

The Baal Shem Tov taught that a descent is the beginning of an ascent, and that the going down is structurally part of the coming up rather than an interruption of it. The Rambam describes exactly this shape without any of that vocabulary. From the head the moon diverges until five degrees, then closes to nothing at the tail, then diverges the other way and closes again. There is no stretch of the pattern that is failure and no stretch that is arrival. Every position on that curve is on the way to the crossing.

Modern Application

What makes the chapter useful outside of astronomy is the discipline of measuring deviation against something you cannot see. The head has no light. It is defined entirely by a relationship, the crossing of one path with another, and it is the only thing in the system that lets you say whether the moon is high or low. Take away the invisible reference and the moon still has a position, still has a constellation, still has degrees and minutes, and there is no longer any such thing as north or south for it.

Most of what a person actually needs to know about himself works this way. Not where he is, which he can usually report, but how far off the plane he is running, which requires a reference that is not itself an object and cannot be pointed at. The Lubavitcher Rebbe returned often to the theme that the Jewish people are compared to the moon, waning and renewing, and that the renewal is not a reward for the waning but the same motion seen from the other end. The measurement in this chapter is not a verdict on the moon. It is a description of where in a known and bounded pattern it currently stands.

The Closing

It is the second of Elul, the month whose whole business is measuring distance honestly. The Sfat Emet taught that the work of these weeks is not the manufacture of a new closeness but the removal of what has settled over a closeness that was never actually lost. That is what a bounded latitude looks like when it is written as a teaching instead of a table. Five degrees is a real distance. It is also a distance with a wall on the far side of it.

And the head keeps moving backward the entire time, three minutes and eleven seconds a day, against the direction of everything else in the sky, so that a man who wants to know where the crossing is this year cannot use last year's answer. The reference point is stable in its rate and never stable in its place. That is why the Rambam makes you compute it fresh, and why he makes you subtract it from a full circle rather than simply reading it off.

Nothing in this chapter can be seen. Not the head, not the tail, not the plane the sun travels in, not the angle between them. What can be seen is a thin curve of light over a field, and two men who will walk into a chamber tomorrow and say so. Everything invisible in the chapter exists to make that visible thing believable. The Rambam is not asking you to look at the moon. He is asking you to know how far off the plane it is running, and to know it well enough that when somebody tells you what they saw, you can recognize the truth in it.