# Algebra for Rebbes

Algebra, like other branches of mathematics, was not something our ancient rabbis took an interest in. It was the Greeks and Arabs who developed it and the Arabs who gave us our name for it, which derives from the Arabic noun jabr. The root meaning of jabr is, curiously, the repair of a broken bone, and its mathematical significance was given it by the ninth-century Baghdadi mathematician Muhammed ibn Musa Al-Khwarizmi in his book Al-kitab al-muh.tasar fi hisab al-jabr wa-l-muqabala, “The Abbreviated Book on the Calculation of the Jabr and the Muqabala.Muqabala means “equivalence” in Arabic, and by jabr Al-Khwarizmi was referring to the “repairing” of a negative quantity on one side of an “equivalence” or equation by replacing it with a positive quantity on the other side — as when, for example, the elementary equation x-8=16 is solved as x=16+8 or 24.

The Passover Haggadah is one of the very few ancient Jewish sources in which even elementary algebra can be found. I have in mind, of course, the passage that we will be reading at the Seder this Monday night in which the Ten Plagues are multiplied, first by Rabbi Yossi, then by Rabbi Eliezer, and finally, by Rabbi Akiva. Rabbi Yossi, as this passage begins, observes that, whereas the third plague, the plague of lice, causes the wizards of Egypt to say to Pharaoh, “This is the finger of God,” at the Red Sea “Israel saw the great hand which the Lord had laid upon the Egyptians.” There being five fingers in a hand, Rabbi Yossi then develops the algebraic equation 1/5 = 10/x, with x standing for the number of plagues at the sea, and arrives at a total of 50.

Rabbi Eliezer, not to be outdone, proceeds to argue that in Egypt there were not 10 but 40 plagues, since the Book of Psalms says of the Egyptians that God “cast upon them the fierceness of His anger: wrath and indignation and trouble, a deputation of evil angels.” If “the fierceness of His anger,” is a term for a single plague, “wrath,” “indignation,” “trouble” and “a deputation of evil angels” tell us that every plague had four manifestations, thus giving us a new equation of 1/5 = 40/x, or 200 plagues at the sea. And Rabbi Akiva then ups the ante even further by punctuating the verse from Psalms differently, changing the colon after “anger” to a comma, so that God casts “upon them the fierceness of His anger, wrath and indignation and trouble, a deputation of evil angels.” This yields a sequence of five parallel terms rather than of four preceded by an inclusive one, and gives us 5 x 10 or 50 plagues in Egypt and 250 on the sea.

You may object at this point that ancient rabbinic Hebrew did not have colons or commas. Yet when it came to the Bible it did have, in the form of the biblical cantillations or chant notes, a system of punctuation that was if anything far more subtle and precise than our own. As uninterested as the rabbis were in mathematics, they were passionately interested in biblical language, which they punctuated not only at key junctures of sentences as we do, but — by means of chant notes having syntactical as well as musical values — at every word. There are 26 of these notes all in all, and they are divided into four groups called “emperors,” “kings,” “ministers” and “servants,” according to how much or little of a sentence they govern.

Is it possible by means of the chant notes to determine who is right about the verse in Psalms, Rabbi Eliezer or Rabbi Akiva? Strictly speaking, the answer is no, since these notes were codified several hundred years after the Mishnaic period in which Eliezer and Akiva lived. But they do tell us which of the two later tradition took to be right — namely, Rabbi Eliezer. This is because the two cantillations over the words “the fierceness of His anger,” h.aron apo in Hebrew, are the ones known as a munah. and a revi’i, the first of which is a “servant” indicating that h.aron is attached to apo, and the second of which is an “emperor” marking the end of a phrase. “The fierceness of His anger” is thus set off from “wrath,” “indignation,” “trouble” and “a deputation of evil angels,” which belong to a separate phrase, very much as though there were a colon between them, just as Eliezer held.

None of this is terribly consequential — indeed, the “algebra” of all three rabbis is, like a great deal of midrashic commentary, more a form of imaginative play than of serious deliberation. Whether there were 50, 200 or 250 plagues at the sea is a matter of pure fancy, since the Bible itself tells us of only one “plague” there: The drowning of Pharaoh’s hosts. Words and not numbers were the rabbis’ strong point.