Tuesday, October 3, 2017

Wall Thickness Between the Tauim

Very few details are provided by the Mishnah for the dimensions of the tauim (small rooms surrounding the Sanctuary Building). In the key to his Temple Diagram, Tiferes Yisrael states what he believes to be the length of the various tauim. Based on his results it is possible to make a reasonable guess as to the thickness of the walls that separated the tauim from each other.

The Facts
Here are some of the facts used in the calculations:
1. There are five tauim on each level along the northern and southern sides. Tiferes Yisrael (Temple Diagram §63) states that the length of these tauim is יד אמות בקרוב, approximately 14 amos.
2. There are three tauim on each of the first two levels in the west. Tiferes Yisrael writes that their length is ז אמות בקרוב, approximately 7 amos.
3. There are two tauim on the third level in the west. Tiferes Yisrael writes that they measure י אמות בקרוב, approximately 10 amos.
4. The tauim are separated from each other by walls with doors in them.

The Assumptions
To make sense of Tiferes Yisrael's numbers, the following assumptions were made:
1. The five tauim of the northern and southern sides must fit into the 73 amos of space between the western wall of the Antechamber and the outer wall of the western tauim.
2. All of the western tauim must fit into the 20 amos of space opposite the Holy of Holies.
3. All of the walls separating the tauim from each other have the same thickness. [This does not include the much thicker walls in the northwest and southwest corners that separate between the western tauim and those in the north and south.]

The Calculations
From the fact that Tiferes Yisrael estimates the length of the tauim it is clear that he performed some sort of calculation to arrive at these numbers. Working backward from his results it appears that his calculations were made by taking the total amount of space available for the tauim of each level, subtracting the amount of space taken up by the dividing walls, and dividing the remainder by the number of tauim.

(S - n × t) ÷ N = L

S = total available space
n = number of dividing walls
t = wall thickness
N = number of tauim
L = tau length

For the northern and southern tauim the known values are:

S = 73 amos
n = 4
N = 5

To pick a wall thickness I looked at the first Mishnah of Bava Basra which lists the various "standard" wall thicknesses used in Talmudic times. The thinnest wall in use back then was 3 handbreadths wide (half an amah), so I set t = 0.5.

(73 - 4 × 0.5) ÷ 5 = 14.2

This value for L agrees nicely with Tiferes Yisrael's "approximately 14 amos."

For the top level in the west, the values are:

S = 20 amos
n = 1
t = 0.5
N = 2

Here L = 9.75 which agrees nicely with Tiferes Yisrael's "approximately 10 amos."

Moving to the first two levels in the west, the values are:

S = 20 amos
n = 2
t = 0.5
N = 3

Here L = 6.33 which, unfortunately, does not quite match Tiferes Yisrael's "approximately 7 amos."

More Assumptions

The results were so good in the first and second calculations that I had to find a way to "fix" that pesky value for the western tauim of 6.33 and get it closer to 7. Here were some of the ideas that came to mind:

1. Perhaps my assumption about the wall thickness is wrong.
2. Perhaps Tiferes Yisrael had a different approach to rounding where any mixed number gets rounded up to the next whole number even when the fraction is less than half.
3. Perhaps there is a typographical error in the text of Tiferes Yisrael's commentary.

I considered #1 briefly but then dismissed it out of hand (obviously).

I had no way of proving #2 (that I could think of) and don't think that it is very likely to be true.

In the end, I decided to use the time-honored solution of changing the girsa of the commentary to match my explanation. My argument was that a very subtle and understandable error crept into the text and that the words ז אמות בקרוב should actually read ו אמות בקרוב (i.e., that the ו was mistakenly printed as a ז). I was so convinced that this must be the case that I set out to find the earliest manuscript or printing that I could get my hands on and hope to find a ו instead of a ז. Bechasdei H-shem I was able to track down a first printing of Tiferes Yisrael on Seder Kodshim and read the original text of his commentary, but that is a story for another post.

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