Friday, December 25, 2009

Comparison of the opinions on the Mishkan


Ok, so here is the analysis of the all interpretations of the dimensions of the Mishkan.

Version "A" (Pentateuch version)

1) In agreement with pentateuch.
2) Very light weight of the resulting Mishkan

Version "B" (Talmud\Rav. Nechemyah version)
1) It is impossible logically and mathematically to confirm stated thickness of the boards(1 Cubit) and stated width of the Holly of Holllies(10 cubits). (as per Talmudic/Rashi description)
2) Stated location of the golden and brass clasps of the first and second coverings of the tent do not make any sense. (both per Pentateuch and Talmud)
3) Unmovable weight of the resulting Mishkan.

Version "C" (Rashi\Baraita version)
1) Location of golden clasps of the first covering directly contradicts Pentateuch.
2) Unmovable weight of the resulting Mishkan.

Version "D" (Talmud\Rav. Yehudah)
1) Corner boards do not fit and do not make any sense (Talmud confirms that)
2) Unmovable weight of the resulting Mishkan.

Wednesday, December 23, 2009

Some comparative calculations - Part 3

Cubit - 0.45 m
Weight of a huge Oxen - 1.5 tonns or 1500 kg.
Oxen pulling load - 1.5 times its own weight
Average density of the goat hair fabric - 1314 kg/m³
Bull skin density - 945 kg/m³
Tahach skin density (will take as an equivalent to bull skin density) - 945 kg/m³
Thickness of 1st, 3rd, 4th covering and Internal Veil - will take equal 14 mm
Thickness of 2nd covering, Outside Veil and Courtyard coverings - will take equal 20 mm

It is also known that the sons of Gershom were given 4 Bulls and 2 Carts. (Numb. 3:25-26)
Thus, we can say that the weight of all coverings and veils was half of the weight of all boards,
pillars and sockets of the Mishkan.

In other words, service of the sons of Gershom has weighted half of the service of the sons of
Merari, which is confirmed by the number of Bulls and Carts given to them for transporting.


Therefore we get:

First Tent covering (with cherubs): Volume - 0.326 m³ - Weight - 429.14 kg (214 kg per half-piece)
Second Tent covering (goat hairs fabric): Volume - 0.511 m³ - Weight - 627.67 kg (314 kg per half-piece)
Third Tent covering (skins of bulls): Volume - 0.384 m³ - Weight - 363.74 kg
Forth Tent covering (skins of tahach): Volume - 0.1119 m³ - Weight - 105.81 kg
Internal Veil (with cherubs кырувами): Volume - 0.0291 m³ - Weight - 38.31 kg
Outter Veril (patterned): Volume - 0.03878 m³ - Weight - 50.96 kg
Sixty (60) courtyard coverings: Volume - 0.5817 m³ - Weight - 764.408 kg (12.74 kg per each covering)

Total weight of all coverings and veils: 2425.06 kg or 2.4 tonn.

Four (4) huge Oxen can transport (1500kg х1.5pulling load)х4 Oxen = 9000kg = 9 tonn.
Whereas a more reasonable value would've been around 4 tonns for 4 bulls and 2 carts(see previous post).

Thus the weight of the coverings and veils of the Tabernacle confirms that, the Tabernacle boards should've been much thinner than commonly accepted Talmudic value of 1 cubit.

Some comparative calculations - Part 2

Ok, so a very good real-life load carrying capacity of a pair of bullocks would be 1.5-2 metric tonns. This includes modern materials and design... not bad.

That should put total load carrying capacity of 8 bullocks at, say, 8 tonns ?!
And Talmud says 56 ??? No, Rav. nehemiah opinion is definitelly wrong here.

Here is an example of what the cart might have looked like:

Taken from here: http://www.patentauction.com/patent.php?nb=4062

Wednesday, December 2, 2009

Yet two more images of the Desert Tabernacle



And thou shalt make a veil of blue, and purple, and scarlet, and fine twined linen; with cherubim the work of the skilful workman shall it be made. And thou shalt hang it upon four(*) pillars of acacia overlaid with gold, their hooks being of gold, upon four sockets of silver. 33 And thou shalt hang up the veil under the clasps, and shalt bring in thither within the veil the ark of the testimony; and the veil shall divide unto you between the holy place and the most holy.
--Exodus 26:31-33
(*) (Only three columns are visible in this image)



And thou shalt make for the screen five pillars of acacia, and overlay them with gold; their hooks shall be of gold; and thou shalt cast five sockets of brass for them. --Exodus 26:37

Two new images of the Mishkan


And thou shalt make a screen for the door of the Tent, of blue, and purple, and scarlet, and fine twined linen, the work of the weaver in colours. And thou shalt make for the screen five pillars of acacia, and overlay them with gold; their hooks shall be of gold; and thou shalt cast five sockets of brass for them. --Exodus 26:36-37


And two boards shalt thou make for the corners of the tabernacle in the hinder part. And they shall be double beneath, and in like manner they shall be complete unto the top thereof unto the first ring; thus shall it be for them both; they shall be for the two corners. --Exodus 26:23-24

Thursday, November 26, 2009

Some comparative calculations















Calculations according Talmud (Rav. Nehemiah opinion):

Shittim Wood = approx. 780 kg/m³ http://www.rmperkins.com/Miscellaneous/Cha...ticsOfBriar.pdf
Talent of Gold = approx. 34 kg
Talent of Silver\Copper - approx. half of the Talent of Gold = 17 kg.
Weight of the very large Ox - 1.5 tonn or 1500 kg
Maximum Wagon Drafting Weight of an Ox - approx. 1.5 times of an Ox weight.
Cubit- approx. 45 cm.
Tent Board Dimensions - 1.5х1х10 cubits or 0.675х0.45х4.5 meters each, out of 48.
Tent Pillar Dimensions - 1х1х10 cubits or 0.45х0.45х4.5 meters each, out of 9
Courtyard Pillars Dimensions - 0.5х0.5х5 cubits or 0.225х0.225х2.25 meters each, out of 60
Total Gold for the Tent: (Exodus 38:24) - will take 20 Talents out of 29.
For transporting all this we have had 8 Oxen (Numbers 3:36 and Numbers 7:8)
Lightweight parts, such as tent bars, pins, ropes e.t.c, are not counted since their weight is negligible.



1) 8 large oxen can draft: (1500kg х1.5draft)х8oxen = 18000kg = 18 tonn total 8 wagon draft weight.
2) Weight of each board of the Tent: (0.675х0.45х4.5)х780 kg/m³ = 1.366875m³ х 780 kg/m³ = 1066.1625 kg.
3) Total Weight of all 48 boards: 48х1066.1625 kg. = 51175.8 kg.
4) Total weight of all 96 sockets for Tent boards: 96х17kg = 1632 kg.
5) Weight of each pillar of the Tent: (0.45х0.45х4.5)х780 kg/m³ = 0.91125m³ х 780 kg/m³ = 710.775 kg
6) Total Weight of all pillars of the Tent: 710.775 kg х 9 = 6396.975 kg.
7) Total Weight of all sockets for the pillars : (4silver х 17 kg.) + (5copper х 17kg) = 153 kg.
8) Weight of each pillar of the Courtyard (0.225х0.225х2.25)х780 kg/m³ = 0.11390625m³ х 780 kg/m³ = 88.846875 kg.
9) Total Weight of all pillars of the Courtyard: 88.846875 kg. х 60 = 5330.8125 kg.
10) Total Weight of gold for the Tent boards: 20х34 kg. = 680 kg.

Total Weight: 51175.8 kg.+1632 kg+6396.975kg+153kg+5330.8125kg.+680 kg. = 564688.5875kg = 56.46 tonn.





Calculations according to my version.


Shittim Wood = approx. 780 kg/m³
Talent of Gold = approx. 34 kg
Talent of Silver\Copper - approx. half of the Talent of Gold = 17 kg.
Weight of the very large Ox - 1.5 tonn or 1500 kg
Maximum Wagon Drafting Weight of an Ox - approx. 1.5 times of an Ox weight.
Cubit- approx. 45 cm.
Tent Board Dimensions - 1.5х0.0225х10 cubits or 0.675х0.01125х4.5 meters each, out of 50(per my description).
Tent Pillar Dimensions - 1.5х0.225х10 cubits or 0.675х0.01125х4.5 meters each, out of 9
Courtyard Pillars Dimensions - 3х0.225х5 cubits or 1.35х0.01125х2.25 meters each, out of 60
Total Gold for the Tent: (Exodus 38:24) - will take 20 Talents out of 29.
For transporting all this we have had 8 Oxen (Numbers 3:36 and Numbers 7:8)
Lightweight parts, such as tent bars, pins, ropes e.t.c, are not counted since their weight is negligible.


1) 8 large oxen can draft: (1500kg х1.5draft)х8oxen = 18000kg = 18 tonn total 8 wagon draft weight.
2) Weight of each board of the Tent: (0.675х0.01125х4.5)х780 kg/m³ = 0.034171875m³ х 780 kg/m³ = 26.6540625 kg.
3) Total Weight of all 50 boards: 50х26.6540625 kg. = 1332.703125 kg.
4) Total weight of all 96 sockets for Tent boards: 96х17kg = 1632 kg.
5) Weight of each pillar of the Tent: (0.675х0.01125х4.5)х780 kg/m³ = 0.034171875m³ х 780 kg/m³ = 26.6540625 kg.
6) Total Weight of all pillars of the Tent: 26.6540625 kg х 9 = 239.8865625 kg.
7) Total Weight of all sockets for the pillars : (4silver х 17 kg.) + (5copper х 17kg) = 153 kg.
8) Weight of each pillar of the Courtyard (1.35х0.01125х2.25)х780 kg/m³ = 0.034171875m³ х 780 kg/m³ = 26.6540625 kg.
9) Total Weight of all pillars of the Courtyard: 26.6540625 kg. х 60 = 1599.24375 kg.
10) Total Weight of gold for the Tent boards: 20х34 kg. = 680 kg.


Total Weight: 1332.703125 kg. + 1632 kg. + 239.8865625 kg. + 153 kg. + 1599.24375kg + 680 kg. = 5636.8334375kg = 5.63 tonn.

So, basically, if the boards had 1 cubit thickness, it would've been impossible to move them on the provided 8 wagons.
Even if we take wood with lower density and assume higher oxen draft, it would still be impossible.

Wednesday, November 18, 2009

A different opinion on the thickness of the boards of the Tabernacle

Here is a different opinion on the matter.

You can find the full original article here
Yeshivat Har Etzion
Parashat TERUMA.
The Structure of the Mishkan.
Thickness of the Mishkan Boards.
by Rav Meir Spiegelman
translated by Kaeren Fish



The length of all the boards of the Mishkan was ten cubits, and their width a
cubit and a half. The Torah makes no mention of their thickness. We must
conclude, then, that this measurement is of no significance. The importance of
the Mishkan is inward rather than outward, and therefore only its internal
measurements are noted.

The internal length of the Mishkan is at least
thirty cubits, since there are twenty boards each a cubit and a half wide. The
width of the Mishkan is at least nine cubits, for there are six boards on its
western side. This number is somewhat inconvenient, for it is not a common
number in the Torah. It is difficult to rest with the assumption that the
Mishkan is thirty cubits long and nine cubits wide. Our difficulty becomes even
more acute in light of the fact that the length of the Kodesh Ha-kodashim is ten
cubits; it does not seem reasonable that its length should be ten cubits if its
width is nine.[4] We are therefore led to conclude that the inner measurement of
width of the Mishkan was ten cubits. This assumption is confirmed by the
structure of the curtains, as will be explained below.



The Sages
determine that the boards were a cubit thick. However, there is some dispute as
to whether they are oblong in shape – such that their thickness remains uniform
throughout their length – or whether they take the form of a split pyramid (a
sort of right-angled triangle), such that their thickness measures a cubit only
at the bottom; at the top they are almost a point. The latter option seems more
probable, for a number of reasons.

First of all, the boards involve a
weight problem. If the board is a cubit thick from top to bottom, and it is made
of solid wood, then the volume of each board is much more than a cubic meter of
wood. Based on a conservative estimate that the specific weight of wood is half
of the weight of water (in fact, different types of wood have different
weights), we conclude that each board weighs more than half a metric ton. Since
the Mishkan included at least fifty boards, we must arrive at the fantastic
total of around twenty-five tons (not including poles, the boards of the screen,
the parokhet, etc.). The Torah states that two wagons served to transport the
boards. Thus, if our calculations are correct, each wagon – drawn by only two
oxen – must have borne more than twelve tons. Clearly, this is not physically
possible, but nowhere is there any indication that the transportation of the
boards in the wagons was in any way miraculous. If, on the other hand, we assume
that the boards stood in the shape of a split pyramid, their weight is halved,
which seems more reasonable.[5]

Secondly, the Torah states that on the
western side of the Mishkan there were six boards, and another two boards for
the edges of the Mishkan. If the boards were a uniform thickness of a cubit and
a half, the boards at the corners should be no different from the others.
Assuming that the internal width of the Mishkan is ten cubits, the width of each
board at a corner could also be a cubit and a half, of which one cubit covers
the width of the boards on the northern or southern side, while the remaining
half-cubit is devoted to the internal width of the Mishkan. Hence there is no
need for the Torah to tell us that there are two boards on the corners. The
general statement telling us that on the western side there were eight identical
boards would have been sufficient. If the thickness of the boards was
negligible, or if they were in pyramid form, then the boards of the corners
would have to be a different shape from the boards in the middle of each side
(in order to close the corner – they would have to be of full thickness), and we
must therefore assert that there were special boards for the corners of the
Mishkan.

Thirdly, the Torah fails to describe how the curtains were
placed upon the boards. The length of the curtains was twenty-eight cubits. This
length was laid sideways upon the Mishkan. The inner width of the Mishkan
measured ten cubits. We are thus left with eighteen cubits. If the thickness of
the boards at the top was a cubit, then each cucovered eight cubits of the
board. This would look strange, for there is no special reason to cover
specifically eight cubits, rather than seven cubits or eight and a half cubits.
The authorities who maintain this opinion would counter that the Torah wanted to
leave one cubit of gold revealed, and – correspondingly – a cubit of silver (on
the assumption that the silver sockets were a cubit high). However, this
hypothesis does not seem to sit well, and it seems preferable to adopt the
opinion according to which the boards were shaped as split pyramids. According
to this view, since the width of the board was small, the curtains covered nine
cubits on each side, down to the height of the sockets (although the outer
length of the boards was a little more than ten cubits, this discrepancy is
negligible).[6] Based on this understanding, the picture that arises seems more
reasonable: the curtains of the Mishkan covered all the gold of the boards. The
goatskin curtains, on the other hand, covered all the boards, including the
silver sockets.

We are left with the question of the need for the
goatskin curtains, and why these cover the silver sockets, while the curtains of
the Mishkan do not cover them.

Saturday, November 14, 2009

Wednesday, November 11, 2009

The Desert Tabernacle and its Corner Boards







1) Western Wall Corner Boards of the Tent



וּשְׁנֵי קְרָשִׁים תַּעֲשֶׂה, לִמְקֻצְעֹת הַמִּשְׁכָּן--בַּיַּרְכָתָיִם
וְיִהְיוּ תֹאֲמִם, מִלְּמַטָּה, וְיַחְדָּו יִהְיוּ תַמִּים עַל-רֹאשׁוֹ, אֶל-הַטַּבַּעַת הָאֶחָת; כֵּן יִהְיֶה לִשְׁנֵיהֶם, לִשְׁנֵי הַמִּקְצֹעֹת יִהְיוּ.
--26 שְׁמוֹת

And two boards shall thou make for the corners of the tabernacle in the hinder part. And two parts of them shall be connected at the bottom, and in like manner two parts of them shall be connected at the top thereof, to form one hollow cylinder; thus shall it be for them both; they shall be for the two corners.
--Exodus 26:23-26:24

Each corner board was made out of two regular boards of the same length, height, thickness and design as the rest of the boards of the tent. These two regular boards were bended into two hollow semi-cylinders and, when connected together, would form one hollow cylinder that would become a corner board. Each such hollow cylindrical corner board would be 10 cubits in height, with outside diameter of 1 cubit, inside diameter of 3/π=0.9549 cubits and thickness of 0.5-(1.5/π)=0.0225 cubits.

Thus, the tent was made out of 48 parts and 50 boards (46 straight boards and 4 curved boards that would form two corner boards)

2) Tents’ Pillars



וְעָשִׂיתָ פָרֹכֶת, תְּכֵלֶת וְאַרְגָּמָן וְתוֹלַעַת שָׁנִי--וְשֵׁשׁ מָשְׁזָר; מַעֲשֵׂה חֹשֵׁב יַעֲשֶׂה אֹתָהּ, כְּרֻבִים.
וְנָתַתָּה אֹתָהּ, עַל-אַרְבָּעָה עַמּוּדֵי שִׁטִּים, מְצֻפִּים זָהָב, וָוֵיהֶם זָהָב--עַל-אַרְבָּעָה, אַדְנֵי-כָסֶף.
שְׁמוֹת --26

And thou shall make a veil of blue, and purple, and scarlet, and fine twined linen; with cherubim the work of the skilful workman shall it be made. And thou shall hang it upon four pillars of acacia overlaid with gold, their hooks being of gold, upon four sockets of silver.
--Exodus 26:31-32

Four pillars inside the Tent were located on the south-north centerline of the 7th boards, counting from the western wall. Pillars were essentially the golden-plated hollow wooden cylinders 10 cubits tall, 0.5 cubits of outside diameter, 1.5/π=0.4774 cubits of inside diameter and 0.5-(1.5/π)=0.0225 cubits thick. Distance between the center of a pillar to the adjacent pillar (or to the wall of the Tent) was 2 cubits. The veil hanged down the pillars into the Ark part of the Tent.

וְעָשִׂיתָ מָסָךְ לְפֶתַח הָאֹהֶל, תְּכֵלֶת וְאַרְגָּמָן וְתוֹלַעַת שָׁנִי וְשֵׁשׁ מָשְׁזָר, מַעֲשֵׂה, רֹקֵם.
וְעָשִׂיתָ לַמָּסָךְ, חֲמִשָּׁה עַמּוּדֵי שִׁטִּים, וְצִפִּיתָ אֹתָם זָהָב, וָוֵיהֶם זָהָב; וְיָצַקְתָּ לָהֶם, חֲמִשָּׁה אַדְנֵי נְחֹשֶׁת.
שְׁמוֹת – 26

And thou shall make a screen for the door of the Tent, of blue, and purple, and scarlet, and fine twined linen, the work of the weaver in colors. And thou shall make for the screen five pillars of acacia, and overlay them with gold; their hooks shall be of gold; and thou shall cast five sockets of brass for them.
-- Exodus 26:36-37

Five pillars were located on the eastern edge, essentially forming the entrance of the Tent. Pillars were the golden-plated hollow cylinders 10 cubits tall, 0.5 cubits of outside diameter, 1.5/π=0.4774 cubits of inside diameter and 0.5-(1.5/π)=0.0225 cubits thick. Distance between the center of a pillar to the adjacent pillar was 2.5 cubits.

The Tent screen door hanged on the pillars outside, facing east.



3) Courtyard Dimensions

אֹרֶךְ הֶחָצֵר מֵאָה בָאַמָּה וְרֹחַב חֲמִשִּׁים בַּחֲמִשִּׁים, וְקֹמָה חָמֵשׁ אַמּוֹת--שֵׁשׁ מָשְׁזָר; וְאַדְנֵיהֶם, נְחֹשֶׁת.
שְׁמוֹת --27

The length of the court shall be a hundred cubits, and the breadth fifty everywhere, and the height five cubits of fine twined linen, and their sockets of brass.
---Exodus 27:18

Dimensions of the courtyard are not given explicitly in the “cubits of length”. Instead they are given in the form of geometry, “cubits of fabric (linen)” and the number of pillars. From the original description and simple mathematical calculations, dimensions of the courtyard come to be 120 cubits by 60 cubits, with 60 hollow wooden cylindrical pillars 5 cubits tall, 0.5-(1.5/p)=0.0225 cubits thick, 1 cubit of outside diameter and 3/p=0.9549 cubits of inside diameter. Each pillar was made out of two equal parts.

Distance between the centers of a pillar to the adjacent pillar was exactly 6 cubits. Each piece of linen that hanged between pillars was 5 cubits wide by 5 cubits long. Connecting silver hanger links between each two pillars were also 5 cubits long each

Tabernacle Layout - Tent and Courtyard

Tabernacle Layout - Tent and Courtyard

Courtyard Elements, South-West Corner


Dimensions of the courtyard are not given explicitly in the “cubits of length”.
Instead they are given in the form of geometry, “cubits of fabric (linen)” and the number of pillars.
From the original description and simple mathematical calculations, dimensions of the courtyard come to be 120 cubits by 60 cubits, with 60 hollow wooden cylindrical pillars 5 cubits tall, 0.5-(1.5/π)=0.0225 cubits thick, 1 cubit of outside diameter and 3/π=0.9549 cubits of inside diameter.
Each pillar was made out of two equal parts.

Distance between the centers of a pillar to the adjacent pillar was exactly 6 cubits.
Each piece of linen that hanged between pillars was 5 cubits wide by 5 cubits long.
Connecting silver hanger links between each two pillars were also 5 cubits long each.

Tabernacle Layout - Tent Dimensions Only


Tabernacle Layout - Tent Dimensions Only

5 Outer Pillars of the Tent


Five pillars were located on the eastern edge, essentially forming the entrance of the Tent. Pillars were the golden-plated hollow cylinders 10 cubits tall, 0.5 cubits of outside diameter, 1.5/π=0.4774 cubits of inside diameter and 0.5-(1.5/π)=0.0225 cubits thick.


Distance between the center of a pillar to the adjacent pillar was 2.5 cubits.

The Tent screen door hanged on the pillars outside, facing east.

4 Inner Pillars of the Tent


Four pillars inside the Tent were located on the south-north centerline of the 7th boards, counting from the western wall. Pillars were essentially the golden-plated hollow wooden cylinders 10 cubits tall, 0.5 cubits of outside diameter, 1.5/π=0.4774 cubits of inside diameter
And 0.5-(1.5/π)=0.0225 cubits thick.


Distance between the center of a pillar to the adjacent pillar (or to the wall of the Tent) was 2 cubits.


The veil hanged down the pillars into the Ark part of the Tent.

Western Wall and Corner Boards


Each corner board was made out of two regular boards of the same length, height, thickness and design as the rest of the boards of the tent. These two regular boards were bended into two hollow semi-cylinders and, when connected together, would form one hollow cylinder that would become a corner board. Each such hollow cylindrical corner board would be 10 cubits in height, with outside diameter of 1 cubit, inside diameter of 3/π=0.9549 cubits and thickness of 0.5-(1.5/π)=0.0225 cubits.


Thus, the tent was made out of 48 parts and 50 boards (46 straight boards and 4 curved boards that would form two corner boards)

Tent Western Wall, South-West Corner Board


Each corner board was made out of two regular boards of the same length, height, thickness
and design as the rest of the boards of the tent. These two regular boards were bended into two
hollow semi-cylinders and, when connected together, would form one hollow cylinder that
would become a corner board. Each such hollow cylindrical corner board would be 10 cubits
in height, with outside diameter of 1 cubit, inside diameter of 3/π=0.9549 cubits and thickness
of 0.5-(1.5/π)=0.0225 cubits.


Thus, the tent was made out of 48 parts and 50 boards (46 straight boards and 4 curved boards
that would form two corner boards)

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