In the end of Parshas Terumah, (pesukim 27:9-19), the Torah describes the beams that held up the curtain that surrounded the courtyard of the Mishkan. Pasuk 10 discusses the beams on the southern side of the courtyard:

In Rashi's seemingly innocent comment on the pasuk, there is a grave arithmetic difficulty which is the subject of much discussion amongst the commentators on Rashi. If there are five

*amos*between each beam and 20 beams, that would provide only 19 spaces of five*amos.*That would yield only 95 of the 100*amos*that the pasuk tells us make up the length of the courtyard. Of course, the first notion is that the space does not include the width of the beams. Therefore, there may have been 95*amos*of space and five*amos*of beams to complete the 100*amos*. This is in fact the suggestion of the Riva, in the name of his rebbe and is also the opinion of the Abarbanel. The 20 beams on the north and south sides added up to five*amos*on either side. This would make each beam one quarter*amah*(1½*tefachim*). This interpretation would avoid all our problems from the outset. However, R' Eliyahu Mizrachi takes issue with this interpretation on two accounts. Firstly, he sees no reason why there should be such a large difference between the thickness of the beams of the courtyard and that of the planks of the Mishkan itself (nine*tefachim*). His second objection is that within the beams themselves you would have some of different thickness than others. On the east and west sides, there are only 10 beams needed to make up five*amos*. (The nine spaces between the ten beams make up 45 of the 50*amos*width of the courtyard.) Therefore, each beam would be three*tefachim*, twice the width of those on the north and south sides. The lack of symmetry involved in this understanding of Rashi causes the Mizrachi to disregard it and give his own interpretation.
Firstly, the Mizrachi suggests that the five

*amos*referred to by Rashi are not five*amos*of space but rather five*amos*from the beginning of one beam to the beginning of the next.. This view is generally accepted amongst all those who deal with this problem with the obvious exception of the aforementioned Riva and Abarbanel. In pasuk 18, the Mizrachi infers from Rashi that the beams were one*amah*thick. Therefore, the actual space between each beam would be four*amos*and the thickness of the beam would complete the five*amos*. However, we have now only accounted for 95*amos*. Therefore, the Mizrachi suggests that the north and south sides actually had 21 beams and the east and west had 11 but that the seemingly extra beam on each side belonged to the set of of beams of the side perpendicular to it. For instance, 21 beams were placed on the southern side of the courtyard. The beam in the southwest corner, though, was officially part of the western side. So, too, the beam in the northwest corner was not counted as part of the western beams but as part of the northern beams and so on. See illustration. With this arrangement another space of five*amos*is added to complete the 100*amos*referred to in the pasuk.
In pasuk 18, the Mizrachi suggests that the 100

*amah*measurement of the courtyard was in fact a measurement from within the beams and the one*amah*taken up by the beams is not included. This reasoning was given in order to justify Rashi's calculation of 20*amos*distance between the Mishkan and the curtains of the courtyard on the north, south and west sides. The Gur Aryeh objects to this with the claim that the pesukim (9,11,12,13) clearly state that the curtains were exactly 100*amos*long on the north and south sides and 50*amos*long on the east and west sides. But according to the Mizrachi's interpretation, the outer perimeter of the courtyard would be 102*amos*by 52*amos*. He offers a defence for the Mizrachi that perhaps the only purpose of the curtains was to cover up the open spaces and they did not need to cover the corners (as illustrated on page 3). However, in his own opinion, the Gur Aryeh suggests that the 100*amah*measurement is in fact referring to the outer perimeter of the courtyard. He then was required to justify Rashi's measurement in pasuk 18 in a different manner.
The Levush HaOrah, another commentator on Rashi is unhappy with both the Mizrachi and the Gur Aryeh's explanations of Rashi in regards to the placement of the beams. From the fact that Rashi mentions the measurement of five

*amos*between each beam more than just once, he infers that Rashi meant for this to be consistent throughout the entire perimeter of the courtyard. According to the Mizrachi the length of the north side, for instance, was really 102*amos*and according to the Gur Aryeh it was 100. However, if you add up 21 beams each of one*amah*thickness and 20 spaces of four*amos*each, we are given 101*amos*. So, too, on the east and west sides we would end up with 51*amos*instead of 50 or 52. He concludes that the only way for the Mizrachi's figures to work out would be to say that one space on each of the four sides was actually one*amah*bigger. For the Gur Aryeh's figure to work one space would have to be one*amah*smaller. The Levush does not accept that such a lack of symmetry was present in the building of the Mishkan and offers a rather unique arrangement of the beams. Each of the beams were circular on the bottom for one*amah*and were inserted into circular holes in the copper sockets that held the beams in place. The beam itself was a semi-cylinder whose diameter was one*amah*. On each of the corners was placed a quarter-cylinder beam so that the curtain could wrap around it. See illustration. The thickness of this beam was only one half*amah*on either side. This removes one half*amah*one either end of each side of the courtyard. With this arrangement, the spaces between all of the beams were all four*amos*wide without any exception and the perimeter of the courtyard was exactly 100*amos*by 50*amos*as stated in the pesukim. Amongst all the interpretations mentioned thus far, this is by far the most symmetric and arithmetically accurate.
Finally, the sefer Ma'ase Choshev offers another possible arrangement of the beams which matches that of the Levush's in symmetry and arithmetic correctness. He suggests that there were no beams in the corners. The curtains were suspended from wooden bars. On these bars were placed the hooks that were used to hang the curtains from the beams. Each of these bars was five

*amos*long. The north and south sides had twenty such bars and the east and west sides had ten. These wooden bars would allowed the curtains to change direction at the corners without the need to wrap it around a beam. See illustration. Once again the figure of five*amos*refers to the distance from the beginning of one beam to the beginning of the next. With this arrangement the thickness of the beams becomes irrelevant. All of the figures mentioned in the pesukim work out perfectly as well. One advantage of this arrangement over that of the Levush's is that all of the beams are the exact same shape.(The illustration assumes the beams to be one*amah*thick.)
The arrangement of the Ma'ase Choshev is the one quoted in the seforim Meleches HaMishkan and Tavnis HaMishkan (etc.). The sefer Lifshuto Shel Rashi, however, is content with the opinion of the Riva and the Abarbanel. Whatever the true arrangement of the beams was, it is clear that when Rashi said that there were five

*amos*between each beam, he had some logical calculation in mind. The only question that remains is "Which?".**On a Related Topic**

The Mishkan was covered by three layers of material(*). The first covering described by the Torah (26:1-6) was made of twisted linen, turquoise, purple and scarlet wool. The covering was made up of 10 panels of 4x28 amos2. This yields a total area of 40x28 amos2. The Mishkan was 30x10 amos2. The beams that made up the walls of the Mishkan were 1 amah thick. Thus, the Mishkan required 32x12 amos2 of roofing.

The beams were 10 amos tall. The covering was 28 amos wide and 12 amos covered the roof of the Mishkan. That leaves 16 amos for the two sides which is 8 amos on each side. So the wool/linen would reach two amos from the ground. There is a dispute as to whether or not the front beams were covered. We will go with the opinion of the gemara (Shabbos 98b) that they were uncovered as Rashi (26:5) notes that the pesukim seem to indicate as such. Therefore, 31 amos of the covering's width provided roofing, leaving 9 amos to hang from the back. The second covering was a covering of goat hair. This covering was wider and longer than the wool/linen layer and covered it fully on all sides.

Rashi (26:13) notes that the Torah teaches us a lesson that one should show compassion for valuable objects. The twisted linen and assorted wools were very precious and thus, as Rabbeinu Bachya explains, it was made not to drag on the ground so that it would not be soiled by dirt and rain and was protected fully by the goat hair. This lesson is easily understood considering the measurements mentioned thus far. However, there is one simple question to be asked. What about the corners? As the accompanying diagram shows, if a piece of material hangs only 8 amos off one side and 9 amos off the other, simple Pythagorean geometry dictates that the corners will hang down more than 12 amos! (This effect is well demonstrated by the corners of a rectangular tablecloth hanging from the table.)This is hardly an efficient way to care for valuables.

This problem seems far too obvious to have been overlooked by Chazal in teaching us this lesson. However, finding the answer was not easy. But finally, an answer was found in R' Chaim Kunyevsky's elucidation of Braisa diMleches haMishkan. There he asks exactly this question. He answers that the corners of the coverings were folded against the back of the Mishkan as illustrated. The Ritv"a (Shabbos 98b) apparently provides the same answer in the name of Braisa diMleches haMishkan but our versions show no evidence of any such discussion. One of the books on the Mishkan actually show such an arrangement but there is no discussion as to any source or reason for it.

*This and a number of other facts discussed on this page are actually subject to a large-scale dispute between R' Yehudah and R' Nechemiah. For our purposes, all figures are according to R' Yehudah.

I wonder... Have seen my blog yet?

ReplyDeletethedeserttabernacle.blogspot.com

Ok, so this is what I think about the dimensions of the Court...

ReplyDeleteIn my humble opinion, the dimensions of the Chatzer of the Mishkan were 120 cubits by 60 cubits.

Here is why:

1) If we would assume that the dimensions of the Chatzer were 100 cubits by 50 cubits, we are going to run into the problems that you have nicely outlined in your post.

And since the traditional interpretations cover almost every concievable permutation of the Court parts that explicitly specified in the original text, one can also conlclude that all of of these interpretations are equaly problematic, and thus - not correct.

2) But how can this be justified? Well, imho it is pretty simple - this is a clear case of misreading of the text.

Exodus 27:9-12 & 14-16 discusses the length of fabric for each side of the court: no problems here. But Exodus 27:13 & 18 seems to be talking about an overall dimensions of the court? Wrong! It also speaks about the length and width of the fabric only!

Thus, the value of 100 cubits refers to the length of fabric on the North and South sides of the court. And the value of 50 cubits refers to the length of fabric on the East and West sides of the court.

But how can we figure out an overall dimensions of the court, if the dimensions of the pillars are not specified explicitly?

Simple! From the dimensions of the Boards of the Tent of the Mishkan!

Each board of the Mishkan was 1.5 cubits long and 10 cubits tall (with an unknown thickness).

Here is an interesting part: if we were to make a tube with these dimensions, taking 1.5 cubits for an inner circumference of the tube, we would get a pillar 10 cubits tall and 1.5/pi=0.477 cubits of inner diameter.

But the court pillars were only half as tall as the boards of the Tent... Well, amazingly, if we will cut the tube that we've calculated above in half and make one wider tube out of these two, we are going to get:

A tube 5 cubits tall, with inner circumference 1.5+1.5=3 cubits, and inner diameter of 3/pi=0.954 cubits. And since this tube obviously had some thickness, the closest whole number would be 1 cubit.

Thus, the each pillar of the court was tube-shaped, 5 cubit tall, 1 cubit in outside diameter and 0.954 cubits of inside diameter, with the thickness of wood of (1-0.954)/2 =~0.0225 cubits (a fingerbreadth).

By doing some more simple math, we get an overall dimensions of the court of the Mishkan as 120 cubits by 60 cubits. (i,e 50 cubits of fabric + 10 pillars one cubit wide each=60 cubits e.t.c)

So I guess Rashi was one of the few who asked the right questions. And of'course, the traditional commentators who identified the pillars of the court being 1 cubit, were also correct, even though they could not explain it and fit all details together.

Sorry ofr the long comment ;)

A very novel idea was put forward by Isaac Hassan of montreal, regarding the dimentions and pillars of the courtyard, which seems to cover all difficulties. It is radical though. The explanation of the Riv"a towards the end of Parshas Terumah seems the most logical, with minor assymetries that can be easily tweaked.

ReplyDeleteI think that we can certainly agree that Mr. Sigalov's explanation is beyond reason.

ReplyDeleteThe explanation of the Riv"a towards the end of Parshas Terumah seems the most logical, with minor assymetries that can be easily tweaked.I would very much enjoy looking at YOUR detailed layout that shows how these "

minor assymetries that can be easily tweaked."Just post link here or in my blog.

Thank you.

Nice post. Question: Has anyone tried to calculate the weight of the aron? There was a lot of gold there! Could it be carried by 4 people "al derech hateva?" Also, when the pasuk says that they used "zahav tahor" does this mean 24 caret gold? Is that practical? Any thoughts? i was inspired to ask this after reading your post on Noach's teiva.

ReplyDeleteHow much did the kapores weigh? And how much gold was donated? This does not include the gold of the aron or keruvim. Nor does it include the menora or other keilim.

Delete1) The dimensions of the kapores in tefachim are 15 x 9 x 1. [There are 6 tefachim in one amah.] The first two dimensions are written in the Torah. For the height [1 tefach] we are relying on Rashi based on Chazal. Using 8 centimeters per tefach it comes to 120 x 72 x 8 or 69,120 cubic centimeters. Pure gold weighs 19.3 grams per cubic cm.

2) Hence the kapores weighed 1,334,016 gr or 1334 kg! [This is more than one ton!!] And this is without the kruvim!

3) The amount of gold donated for the Mishkan [see Shmos 38:24] was 29 kikar and 730 shekel using shekel hakodesh. Rashi tells us that 1 kikar equaled 60 maneh [מנה]. However, the maneh of kodesh was doubled, so there were 120 maneh in 1 kikar. One maneh equals 25 sela'im [the Talmudic term for the Torah shekel]. Hence, 1 kikar is equal to 3000 shekalim.

4) So what was the weight of these donations?

5) The donation was [29 times 3000 + 730] 87,730 shekalim. This equals [87,730 times 14 gram] 1228.22 kg.

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