## Sunday, April 23, 2006

### Measures of Reviit in Bavli vs. Yerushalmi

The measure of reviit is different in Bavli and Yerushalmi.
Bavli states it is 2 X 2 X (2 + 1/2 + 2/3) fingerbreadths.
Yerushalmi states it is 2 X 2 X (1 + 1/2 + 1/3) fingerbreadths.

While Bavli is straightforward to figure out, Yerushalmi is not so obvious. What follows is my suggestion, which is simply that the ambiguity lies in the 2 possible measures of cubit.

To cite Rif:

Rav Chisda said: The Biblical reviit is 2 X 2 X 2 7/10 fingerbreadths.
And measured with the thumb, as they learnt {in a brayta}: {Vayikra 15:16}:
 טז וְאִישׁ, כִּי-תֵצֵא מִמֶּנּוּ שִׁכְבַת-זָרַע--וְרָחַץ בַּמַּיִם אֶת-כָּל-בְּשָׂרוֹ, וְטָמֵא עַד-הָעָרֶב. 16 And if the flow of seed go out from a man, then he shall bathe all his flesh in water, and be unclean until the even.
In water = the water of a mikveh
All of his flesh in water = water into which all of his body may enter.
And how much is that? 1 cubit X 1 cubit X 3 cubits high. {That is, three cubic cubits.} And the Sages gave the measure of the waters of a mikveh to be 40 seah.

To explain: The Biblical reviit is 1 1/2 eggs volumes. And the log is 4 reviit which are 6 eggs. {Since 4 X 1.5 = 6.}
And how do we know that a log is 4 reviit? For they learnt {in the Mishna in Menachot 87b that there are 7 liquid measures in the Temple}: The hin, 1/2 hin, 1/3 hin, 1/4 hin, the log, 1/2 log, 1/3 log, 1/4 log. {The Mishna which lists 7 skips 1/3 log, which for some reason is listed here.}

Thus the log is composed of 4 reviit.
The hin is 12 log, for it is written זה, and זה is in gematria 12 log.
Thus, a hin is 48 reviit {because 12 X 4 = 48}, which are 72 eggs {because 48 X 1.5 = 72}.
A kav is 4 log which is 16 reviit {because 4 log per kav X 4 reviit per log = 16} which are 24 eggs {because 16 X 1.5 = 24}.

And how do we know that the kav is 4 log? For they learnt {in a Mishna}:
Hillel says: A hin full of drawn water invalidates the mikveh - for one is obligated to relate {the teaching} in his teacher's language - and how much is a hin - 12 log.
Shammai says: 9 kav which are 36 log.
Thus, the kav is 4 log {since 9 X 4 = 36}.

The seah is 6 kav which are 2 hin {since 6 kav = 24 log, and there are 12 log to a hin} which are 24 log {since there are 12 log to a hin} which are 96 reviit {since 24 X 4 = 96} which are 144 eggs {since 96 X 1.5 = 144}.

The eifah is 3 seah, which are 18 kav {since there are 6 kav to a seah} which are 6 hin, which are 72 log, which are 288 reviit, which are 432 eggs.

The measure of challah is an omer, which is 1/10th of an eifah, which is 43 eggs. And so too for matzah.

The Biblical reviit is 2 X 2 {X 2 7/10} fingerbreadths.
{How so?}
The Sages measured the water of a mikveh as 40 seah. The seah is 6 kav. The kav is 4 log. The log is 4 reviit.

Now, 1 cubit X 1 cubit X 3 cubits high {which is the dimensions of a mikveh} contains 40 seah.

We can do a simple calculation here, easier than that of the Rif. Thus, at this point I divert from translating the Rif. Since to convert cubits to fingerbreadths, we multiply by 24 (that is, X 6 X 4), we may calculate that:
1 cubic cubit = 24 X 24 X 24 fingers = 13824 cubic fingers.
3 cubic cubits = 3 X 13824 = 41,472 cubic fingers.

This is for 40 seah. Divide by 40, then by 6, then by 4, then by 4 again to get 1 reviit.
Thus, the ratio of 40 seah to 1 reviit is 3840:1.
Thus, 41,472 / 3840 will give us the cubic fingers for 1 reviit.
41,472 / 3840 = 10.8 cubic fingers.

Is this 2 X 2 X 2 7/10 fingers?
2 X 2 X 2.7 = 10.8 exactly.

However, Rif does not have modern geometry, and so he must resort to a more complicated calculation to acheive the same result. Feel free to skip over that, since it is after all just math.

Now, the Yerushalmi gives a different volume measure for a reviit, namely 2 X 2 X 1 + 1/2 + 1/3.
How can we arrive at that amount. I would posit the following - the only obvious room for ambiguity here is in the amah - the cubit. There were two cubit measures. One was composed of 6 handbreadths and the other of 5 handbreadths. The reviit of Bavli is computed using the 6 handbreadth cubit. Let us recalculate assuming a 5 handbreadth cubit, and see if we can arrive at the measure described in the Yerushalmi.

Since to convert cubits to fingerbreadths, we multiply by 20 (that is, X 5 X 4), we may calculate that:
1 cubic cubit = 20 X 20 X 20 fingers = 8000 cubic fingers.
3 cubic cubits = 3 X 8000 = 24,000 cubic fingers.
This is for 40 seah. Divide by 40, then by 6, then by 4, then by 4 again to get 1 reviit.
Thus, the ratio of 40 seah to 1 reviit is 3840:1.
Thus, 24,000 / 3840 will give us the cubic fingers for 1 reviit.
24,000 / 3840 = 6.25 cubic fingers.

How much is that?
That is 2 X 2 X 1.5625 fingerbreadths.
It is 1 + 1/2 + 1/16 fingerbreadths.

This is only slightly less than the Yerushalmi's measure of 1 + 1/2 + 1/3 fingerbreadths.

However, realize first of all that they were not using modern methods of calculation, and so to come to a figure with such precision would be quite difficult. Furthermore, the way these measures seem to work in both Bavli and in Yerushalmi is as the sum of a series of fractions.

Thus, in Bavli, the height is calculated as 2 + 1/2 + 1/5.
And thus, in Yerushalmi, the height is calculated as 1 + 1/2 + 1/3.

In theory, we might have imagined a series of continuously reducing fractions (with zero as the numerator until we reached 1/16), but such precision is unnecessary and overly complicated. Instead, the "good enough" way of describing numbers might be to continuously add smaller fractions until you have the minimum with possibly a bit over. We could not add 0/3 to 1/2 because we would have too little; adding 1/3 takes us over the top, and is thus sufficient. 