The Doubled Distance

The Hook

Everything in the last several chapters has been average. The mean position of the sun. The mean position of the moon. The mean of the moon within its own path. Averages are what you get when you assume nothing ever varies, and the Rambam has been perfectly candid that nothing up there varies at all, that each body proceeds at a uniform speed and never inclines toward heaviness or lightness. The unevenness is ours. It comes from measuring a circle from a point that is not its center.

Chapter fifteen is where he closes the gap. And the instrument he reaches for is not a better measurement of the moon. It is the distance between the moon and the sun, doubled. Where the moon truly is turns out to be a question you cannot answer by looking at the moon.

Double the Elongation

The procedure opens without preamble. If you wish to know the true position of the moon on a given date, calculate the mean of the moon at the time of the sighting, then the mean of the moon within its path, then the mean position of the sun. Subtract the sun's mean from the moon's mean. Double what remains. That figure has a name: the double elongation.

Then a sentence that quietly reminds you what all of this is for. The intent of every calculation in these chapters, he says, is to know how to sight the moon. Not to describe the heavens. To decide whether a court may accept witnesses tonight. And in that context the size of the double elongation matters, because it is impossible for it to be less than five degrees or more than sixty-two degrees on the night the moon is to be sighted. Its measure will never exceed or fall short of those numbers.

Read that as the working astronomer's sentence it is. He has just handed you a check on your own arithmetic. If your double elongation comes out at three, or at eighty, you have not discovered a strange month. You have made a mistake, and you now know it before the court does. The bounds are not decoration. They are the guardrail on a night when a wrong answer means a wrong festival.

What the double elongation buys is a correction to the mean of the moon within its path. Five degrees or thereabouts calls for no increase at all. Between six and eleven, add one degree. Between twelve and eighteen, two. Between nineteen and twenty-four, three. Between twenty-five and thirty-one, four. Between thirty-two and thirty-eight, five. Between thirty-nine and forty-five, six. Between forty-six and fifty-one, seven. Between fifty-two and fifty-nine, eight. Between sixty and sixty-three, nine. The figure that emerges after the addition has its own name: the correct course.

Look at the shape of that table. The farther the moon has pulled away from the sun, the larger the correction the reckoning demands. The relationship is not incidental and it is not decorative. The moon's apparent misbehavior is measured entirely by its separation from the light it borrows from, and the further out it has traveled the more the average has to be adjusted before it will tell you the truth.

The Angle That Grows and Then Shrinks

With the correct course in hand there is one more quantity, the angle of the course, and one decision about what to do with it. If the correct course is less than 180 degrees, the angle is subtracted from the mean of the moon at the time of the sighting. If it is more than 180 degrees, the angle is added. What remains after the addition or the subtraction is the true position of the moon at the time of sighting.

That single sentence carries more weight than anything else in the chapter, and it is easy to read past it. Crossing 180 degrees does not change how large the correction is. It reverses which way it points. A reckoning that is perfect in every figure and blind to that threshold will place the moon on the wrong side of itself by twice the angle, and every number on the page will still look reasonable.

And at exactly 180 degrees, or at an even 360, there is no angle at all. The mean position of the moon at the time of sighting simply is its true position. At those two points the average tells the truth without help.

Then the table of angles, and it does something worth staring at. A course of ten degrees yields an angle of fifty minutes. Twenty yields one degree and thirty-eight minutes. Thirty, two degrees and twenty-four. Forty, three degrees and six. Fifty, three degrees and forty-four. Sixty, four degrees and sixteen. Seventy, four degrees and forty-one. Eighty, an even five degrees. Ninety, five degrees and five minutes. One hundred, five degrees and eight minutes. And then it turns around. One hundred and ten, four degrees and fifty-nine. One hundred and twenty, four degrees and forty. One hundred and thirty, four degrees and eleven. One hundred and forty, three degrees and thirty-three. One hundred and fifty, two degrees and forty-eight. One hundred and sixty, one degree and fifty-six. One hundred and seventy, fifty-nine minutes. At an even one hundred and eighty, nothing.

The correction climbs to its maximum at a course of one hundred degrees, five degrees and eight minutes, and then falls away to nothing. The gap between where the moon appears to be on average and where it truly is is widest in the middle of the arc and vanishes at the ends. If the course runs past 180, you subtract it from 360 to find its angle, exactly as with the sun. If it carries units as well as tens, you take the average increase per degree and add the proportionate share to the lower figure, again exactly as with the sun.

Friday Night, the Second of Iyar

He will not leave it abstract. Suppose we want the true position of the moon on Friday night, the second of Iyar, in the year that serves as the starting point, twenty-nine complete days after the epoch. The sun's mean is 35 degrees, 38 minutes, 33 seconds. The moon's mean at the time of sighting is 53 degrees, 36 minutes, 39 seconds. The mean of the moon within its path is 103 degrees, 21 minutes, 46 seconds.

Subtract the sun's mean from the moon's mean and the elongation is 17 degrees, 58 minutes, 6 seconds. Double it: 35 degrees, 56 minutes, 12 seconds. That falls between thirty-two and thirty-eight, so five degrees are added to the course, which makes the correct course 108 degrees and 21 minutes. And then a line easy to skip and impossible to unsee once you have seen it: as with the sun, the minutes are of no consequence in the calculation of the course. He has been carrying seconds through five operations and here he drops twenty-one minutes without a second thought, because he knows exactly which precision buys something and which is theater.

The angle for a course of 108 is 5 degrees and 1 minute. The correct course is less than 180, so it is subtracted from the moon's mean, leaving 48 degrees, 35 minutes, 39 seconds. The seconds are rounded off and counted as a minute. The true position of the moon that night is 48 degrees and 36 minutes, which is 18 degrees and 36 minutes into Taurus, in the nineteenth degree of that constellation. And with that, he says, you can find the true position of the moon at the time of sighting for any date you like, from the beginning of this starting year until the end of all time.

The Unifying Principle

Nothing in this chapter measures the moon by itself. Every figure that finally locates it is a difference, a doubling of a difference, or a correction derived from that doubling. The moon does not have a position in the sense we casually mean. It has a relationship, and its position is what the relationship works out to on a given night.

Chassidus has never been shy about which body is which. The moon is the receiver, the one with no light of its own, the one whose whole visible life is a function of where it is standing relative to a light it does not produce. The Alter Rebbe writes at length about a soul whose entire radiance is borrowed and continuous, illuminated not because of anything it manufactures but because of what it faces. Chapter fifteen is that idea as arithmetic. You cannot compute the moon from the moon. Subtract the sun and double it, and only then do you know where the moon is.

There is a second teaching hiding in the table of angles. The Baal Shem Tov's principle of individual providence is usually heard as a claim about scope, that nothing is too small to be watched. It is also a claim about precision, that every particular thing is placed exactly where it belongs. And here the correction is not uniform, not a flat allowance applied to every night. It is a different figure for every position, largest in the middle and nothing at the turning points. The distance between your average and your truth is not a constant. It depends on precisely where you are standing.

The Rebbe spoke often about the difference between what a person is on average and what a person actually is, and refused to let anyone settle for the average as a self-description. The mean position is a fiction of enormous usefulness. It assumes nothing ever varies and it is wrong on almost every night of the year. But it is not discarded. It is the only thing you have to correct from. The Rambam does not throw out the mean and start over. He computes it faithfully, and then he asks how far the moon stands from the sun, and the answer to that question is what turns the fiction into a fact.

Modern Application

We are fluent in mean positions. The average week. The way things generally go. The person we usually are, which is the figure we quote to ourselves when we want a quick answer and the figure we resent when somebody else quotes it back. It is a real number and it was honestly computed, and on any given night it is off, and the chapter is very specific about what it would take to correct it.

It would take knowing how far you are standing from the light. Not how you feel, not how the week went. The measured distance. And the correction that follows is not a matter of taste; it is fixed by a table, and it is bigger the farther out you have drifted, and above all it can point in either direction. On one side of the threshold you are further along than the average said. On the other side you are behind it. The same size of correction, applied the wrong way, doubles the error and looks entirely respectable on the page.

There is a last consolation in the shape of the angles. The gap between the average and the truth is at its widest in the middle of the arc, and it closes to nothing at the turning points. The times you feel least legible to yourself, when no account of who you are seems to fit, are the middle of a course, which is exactly where the correction is largest and where the mean is least worth trusting. At the turning points, where something is beginning or ending, the two agree.

The Closing

All of this arithmetic exists for one purpose: so that a court sitting on the thirtieth of the month knows whether tonight the moon can be seen. Not where it is in some absolute registry of the heavens. Whether a human being standing in a field outside Jerusalem could catch it, and come, and testify. The mathematics is in the service of a sighting, and the sighting is in the service of a month beginning.

Today is the first of Elul, and it arrived because somebody once did this work. The moon has no light of its own and the reckoning that finds it does not ask it to have any. It asks only how far it stands from the sun, doubles the answer, and corrects from there. The mean position was never a verdict. It was the starting figure, and the entire chapter is the Rambam telling you not to mistake one for the other.