Number π and Solomon's molten sea
As
stated in Ch.7 of the book of Kings I, King Solomon ordered Hiram from Zor to make, among other things, a molten sea. "And he
made a molten sea, ten cubits from the one brim to the other: it was round all
about, and its height was five cubits: and a line of thirty cubits did circle
it round about" (Kings I, Ch. 7:23) (see the side view and the view from above). Since the diameter of the vessel was 10
cubits and the circumference 30, their ratio was 3.
Quite a poor approximation of the number π=3.1415…! Perhaps the diameter
of 10 cubits included the thickness of the walls of the vessel, while the
perimeter was measured from inside? Indeed the Talmud in Iruvin
14a makes such a suggestion. However it is rejected
immediately. The Talmud cites the verse 7:26
which says "and it was a hand breadth thick, and brim was wrought like the
brim of a cup, like the petals of a lily".
The Talmud infers from this sentence that the thickness of the brim was
negligible. Besides, the Talmud claims that both measures: the diameter
and the perimeter are the inner ones. The Talmud comes from it to the
conclusion that the ratio of circle to its diameter is 3! In the sequel the
Talmud discusses the volume of the vessel. At the end of the verse
7:26 is written: "it
contained two thousand bat". The "bat" equals 3 seah while 40 seah make up a miqveh of 3 cubic cubits. Hence the volume of the sea was
450 cubic cubits. The Talmud however calculates the volume of the cylinder
assuming that π=3 and arrives at 3∙52∙5=375 cub. cubits, i.e. 5/6
times less than required. Then Talmud
quotes an external source ("braita") saying
that the Solomon sea for the first 3 cubits of its
height was square and for the next 2 cubits was round. Hence the volume of the
sea was 102∙3+3∙52∙2=450, exactly as
stated in the book of Kings!
Is it possible that the sages of Talmud
thought that π is exactly 3? Even a rough measurement would show that
π is larger than 3! The approximation of π by the ratio 22/7 was
known in the ancient time. It is impossible to imagine that King Solomon, the
wisest of all men, did not know this elementary fact. And at the same time he
was able to manufacture a gigantic copper vessel of a given size and
volume.
Somebody drew attention to the fact
that the word "line" in the verse 7:23 is written in a strange way (some attribute this
remark to Hagra- the genius from Vilno,
but in his books it is not found). Namely. it is
written as ÷åä (Quf,
Vav, Hey) instead of the correct ֌, while on the
margin of the page appears a correction: read ֌ (Quf,Vav). There
are several instances in the Tanach when a word is
spelled in one way and is pronounced a little differently. As a rule we don't
know the reason for these differences. In our specific case the following
explanation was suggested. As well known, the Hebrew letters have numerical
values, Alef=1, Bet=2, Gimmel=3,
Dalet=4, Hey=5, Vav=6, etc, Yud=10, Qaf=20
Lamed=30 etc., Quf=100, Reish=200
etc. A word has a numerical value (Gematria) equal
the sum of its letters. Hence the word ÷åä
(qava) has a numerical value of 100+6+5=111 while the
word ֌ (qav) - 100+6=106. If
one multiplies the "wrong" ratio 30/10 written explicitly by the
correction ÷å/÷åä=111/106, one obtains
the ratio 333/106=3.141509. This is an approximation of π=3.1415926… with
5 significant digits.
The number
π as any number could be expanded into continuous fraction a+1/(b+1/(c+1/(d+…)). In the
case of π a=3, b=7, c=15, d=1. The ratio 333/106=3+(1/(7+1/15))
is a second approximation of π. The fist approximation is 22/7, the third
is 355/113=3.1415929… It is remarkable that the first three approximations of
π are extremely efficient, i.e. they are very close to π while having small denominators.
It
appears to me that the correction ÷å/÷åä (qava/qav) has not merely
numerical meaning. The word ÷åä (qava) is feminine (in
Hebrew the feminine words almost always end with ä) while ÷å (qav) is
masculine. The way the word is spelled is called "masoret"-îñåøú and is
feminine, the way it is pronounced is called "mickra"-
î÷øà and
is masculine. On the other side, in the pair circle-diameter, the circle
represents a feminine, material notion (e.g. the mother Earth) while the
straight line represents the masculine, spiritual notion (e.g. the rain that
fertilizes the earth). Hence the word ÷åä (qava) is
related to the circle while ֌ (qav) to the diameter. With
this correspondence the
verse 7:23
reads "֌
(qav) ten cubits from the one brim to the other … and
a ÷åä (qava) of thirty
cubits did circle it round about". Thus the ratio of a circle to diameter
becomes (30xqava)/(10xqav)=333/106.
Notice that that all objects in the tabernacle where
straight. May be this is the reason why Rambam draw
the Menorah with
straight branches? If in the "heavens", in the spiritual world, there
are no curved lines, the circle is perhaps represented there by a polygon. In
case of the perimeter of the circle, the hexagon could serve as a model. In
case of the area, the dodecagon could be the model. In the first case the
perimeter is equal 2∙3∙radius of the surrounding circle; in the
second case the area equals 3∙square of the radius, as if π=3. That
is why the Sages considered the equality π=3 not as an acceptable
approximation but as a reflection of a certain spiritual truth.
The idea that π represents the ratio of feminine to
masculine could be demonstrated by the number 135. Doubling of its digits
113355 and separation 113:355 leads to the ratio 355/113 that approximates
π with seven significant digits!
The analogy is the double faced Adam and his separation into a man and a
woman. The same idea could be traced in the volume of kaporet-
the cover of the arc. Its length was 2.5 cubits or 15 palms, the width was 1.5
cubits or 9 palms and the thickness was one palm. Thus
its volume was 135 cub. palms. Two cherubim, a boy and
a girl, where growing out of it, separated by a tiny gap between their wings.
The same was the volume of the sockets under the pillars of the tabernacle
(their dimension was 4.5x6x6 palms from which one should subtract the volume of
the wooden legs of the pillars 1.5x3x6 palms). Two sockets were put on two legs
of a pillar and were united by the pillar. We encounter again the number 135 in
the plan of the Temple
Mount. According to the
Mishna Middor (Measures) ch.
2, the width of the inner men's court and of the width and length of the
adjacent women's court was 135 cubits. The idea of approximation of π by
the ratio 355/113 could be traced also in the holy language. The number 355 is
the gematria of the word ùðä (Shin Nun Hey)-the year. The length of the
12 month lunar year is also 354-355 days. The yearly cycle represents the
circle. The word ùðä also means repetition, as returning back around the circle. The
number 113 is the gematria of the word ôìâ (Pey, Lamed, Gimmel), which means
half or division into parts. The same is the gematria
of the word çöéä (Chet, Zadi,Yud,
Hey) – halving. Indeed, diameter divides the circle into two equal parts. Thus
the fraction 355/113 represents the ratio of the circle to the diameter.
Let us discuss now the volume of the molten sea. As already
mentioned above, this volume according to Kings I 7:26 was 2000 bat or 450 cub. cubits. However, the volume of the cylinder of the diameter
10 and the height 5 cubits is π52∙5=392.7 cub. cubits. If one assumes according to the Talmud Eruvin 14a that the lower three cubits of the sea were a
square of 10 by 10 and the upper two cubits were a circle with diameter 10,
then one obtains the volume 300+ π52∙2=457.08 cub. cubits. One can suggest that the Scriptures rounded down
this number to 450, or adjusted it to the approximation of π by 3. But, since a hidden deep meaning was found in
the perimeter of the sea, we will try to find such meaning also in the volume
of the sea. For that sake one should
first understand by what cubits the sea was measured. It is written in Talmud Iruvin 83a that the unit of volume "seah" (equal to 3/40 cub. cubits) increased in Jerusalem by a factor
6/5. Talmud does not mention that the length of the cubit increased too. The generally accepted length of the cubit of
Torah is 48 cm (so called measure of Rabbi Haim Nae).
"Competing" with it is so called measure of Hazon
Yish which is 6/5 times longer. Lately, by studying the length of the cubit
mentioned in the book of Ezekiel (Ch. 40) and its relation to the
size of the Land
of Israel, I came to a
conclusion that this cubit is exactly 51 cm (see the abstract and other
material on the site www.truthofland.co.il
in Hebrew and English). Since, by
Ezekiel, the altar in the Third Temple will be measured by this cubit and since
(according to Rambam) the size of the future altar
will be the same as it was in the Second Temple, we infer that the Second (and
hence the first) Temple was measured by the cubit of 51 cm. Notice that the
increase of the cubit by factor 51/48 results in the increase of the volume by
factor (51/48)3=1.19946... . For practical matters this number is
equal to the ratio 6/5=1.2 between the unit of volume in Jerusalem and the unit of volume of Torah. We
are driven to a following conclusion: King Solomon has increased the length of
the cubit from 48 to 51 cm and accordingly increased the unit of the volume by
factor 6/5. Why then the Talmud does not mention the increase of the cubit?
Because this increase does not fit exactly the increase of the volume!
Nevertheless, the Sages mention in the Midrash Rabba
31,10 that the Temple
was measured by a special cubit used by Noah in his Ark. Now we understand the meaning of the
cubit of Hazon Yish. The above mentioned Talmud Iruvin 83a
says that as the Sanhedrin moved to the town of Zippory, the Sages
increased the unit of volume of Jerusalem
by factor 6/5 from. Let us add that in our time, under the influence of Hazon Yish, the unit of volume in
Bnei Brak was increased
once again by factor 6/5. The threefold increase of the volume by factor 6/5 is
equivalent to a one time increase of the cubit by the same factor. This is how
the cubit of 6/5x48 cm appeared.
After this discourse let us return to the sea of Solomon.
Since the Temple
was measured by the cubit of 51 cm, so was the molten sea. On the other hand,
the volume of the sea of 2000 bat was given in the units of Torah. Hence this
volume was 6/5 bigger than the volume of the cylinder of diameter 10 and a
height 5 cubits, in full agreement with the text of the Scriptures. However, this agreement is based on the
approximation of π by 3 and of (51/48)3 by 1.2. If we use the
correct value of π and of (51/48)3 then the volume of the sea
will be 1.04673 times bigger. Let us return to the description of the walls of
the sea in the verse 7:26 "and it was a hand breadth thick, and brim was
wrought like the brim of a cup, like the petals of a lily". We will
assume that the sea was a cylinder of diameter 10 cubits from outside so
that the thickness of the walls decreased its inner volume. The wall was hand
breadth (1/6 of the cubit) thick at the bottom of the sea and practically zero
at the top. What is the form of lily? Certainly not straight. The simplest
curve next to the straight line is an arc of a circle. It is natural to assume
that this arc was perpendicular to the bottom of the sea (see the side view). It is not difficult to calculate the volume of such
wall. Denote by r the radius of the exterior cylinder (r=5), by h its height
(h=5) and by d the thickness of the wall at the bottom of the cylinder (d=1/6).
The radius of the arc is R=(h2+d2)/(2d)=75.0833.
The wall is the body of rotation around the axis of the cylinder. Its volume is
V1=2π∫(R+r-d-x)(R2-x2)1/2dx, R-d≤x≤R
=2π((R+r-d)(R2arcsin(h/R)-(R-d)h)/2-h3/3)=17.224386.
The volume of the sea is
V=πr2h-V1=375.474696 cub. cubits
of 51 cm. If we translate them into cubits of 48 cm, we obtain 450.36796 cub. cubits. It is almost the requested number. But we did not
pay attention to the fixtures of the sea. The verse 7:24 reads "and under
the brim of it round about there were knops (peqaim
in Hebrew) compassing it, ten in a cubit, compassing the sea round about; the
knops were in two rows, cast with it in the same casting" (see the view from above). Perhaps
these knops were cast in the walls from inside and thus decreased the volume of
the sea? The word peqaat in Hebrew means something
round, apparently a ball. If ten of them were in one cubit then the diameter of
each one was 1/10 cubit. Interestingly enough, the ancient Aramaic translation
of Jonathan ben Uziel calls these peqaim- eggs. By definition, the volume of an egg is 1/1920
of a cub. cubit. The volume of a ball of diameter 1/10
is 4/3π(1/20)3=1/1909.86.
If one approximates π by the ratio 25/8=3.125 then one obtains exactly
1/1920. How many balls were in a row? If we calculate the perimeter of the brim
by the correct value of π, we obtain 314.5 balls. Most likely the number
of balls was "round", 300. The perimeter of 30 cubits as stated by
the verse 7:23 could hint
to this number of balls. There is no contradiction between this number of balls
and the actual perimeter- there were small gaps between the balls. Two rows of
balls – total of 600. However, since the balls were immersed partially into the
thickness of the walls, we should subtract the volume of the immersed part. We
will assume that the upper row of balls was sitting just below the brim and the
second row right below the first one (see the side view). Then the
center of the balls of the first row lies 1/20 cubit below the brim and of the
second row 3/20 cubits below the brim. The thickness of the wall there is
correspondingly 0.0033 and 0.0099 of a cubit. The immersed volume of the first
row is equal approximately to a volume of one ball and of the second row- about
8.2 balls. High accuracy numerical integration gives respectfully 1.03 and
8.30, together 9.33 balls. Hence we should subtract from the volume of the sea
590.67 balls. With a volume of a ball 1/1909.86 cubic cubits,
we obtain the volume of the sea 375.165422 cub. cubits
of 51 cm or 449.99700 cub. cubits of 48 cm. If we
count the ball as an egg of 1/1920 cub. cubits, we
obtain the net volume of the sea 449.999 instead of 450.0. A relative error of
1/500,000! Of course, this is a
theoretical accuracy since a copper vessel would expand under the pressure of
water and even the water would compress under its own pressure.
Let us summarize the results of our investigation. Obviously,
King Solomon knew not only the value of π with high accuracy but was also
able to calculate the volume of a body of rotation with 6 digits. But we are
talking here not about human wisdom but of divine wisdom. No human mind could
design a vessel of such simplicity and harmony and at the same time arrive at a
"round" volume with 6-digit accuracy. Also the play of words with the
line ÷åä/÷å (qava/qav) which corresponds to
the approximation of π by a continuous fraction, should have been built in
the Hebrew language.
Our story would be incomplete if we will not bring the
reaction of critics. What do they say? The pair of words ÷å/÷åä (qava/qav) appears in the Scriptures in two more places: in Jeremiah
31:38 and Zachariah 1:16. In both cases the text does not talk about the length
of a circle and the number π. Hence the difference between the spelling
and the reading of this word does not have the meaning we attributed to it.
What can we reply? The same word or group of words could be used in different
places to code different things. The fact that we don't understand the coding
in one place does not diminish our understanding in another place. Yet, I will
suggest an interpretation of this pair of words in the above two places. In
both cases the text talks about the future expansion of Jerusalem. This expansion is described in
detail in Ezekiel Ch. 45 and Ch. 48. The future Jerusalem, more precisely the oblation (truma) from all the tribes will be a square of 25,000 by
25,000 measuring rods (see there the verse 48:20 and the map).
Since the measuring rod is 6 cubits long, the size of the oblation is 75 by 75
mils (one mil equals 2000 cubits; according to my calculation it is 1.02 km).
This square will be divided in three parts. The northern 30 mils will belong to
the priests (cohanim), the next 30 mils will belong
to Levites and the southern 15 mils will belong to the servants of the city.
The Temple will
be in the middle of the part of the priests. To the west and to the east of
this square will extend the domain of the King-"Nassi".
To the north of the square will be the land of seven tribes and to the south of
it –the land of five tribes. Each tribe will be allocated a strip of land of
the same width: 75 mils from north to south. With additional information (see abstract § 11) it
was possible to reconstruct also the western and eastern borders. It turned out
that the land allocated to 12 tribes is exactly 400 by 400 mils. However, the
Talmud says in several places (e.g. Megila 3a) that
the territory of Israel is 400 by 400 parsa,
where parsa is 4 mils. Thus the territory of Israel
should be 16 times bigger. We did not count yet the area of the strip of the
King. Ezekiel does not say how far it extends to the west and east. Could it be
that this strip, which is 75 mils from south to north, surrounds the whole world? The Talmud (Chulin
91b) says that the ladder of Jacob with the angels ascending and descending on it, was 8000 parsa wide (two
angles ascending and two descending, each angel 2000 parsa
wide). Regarding the length of the ladder, Rashi (the
main commentator of the Torah) writes in Genesis 28:17 that the basis of the
ladder stood in Beer Sheba while its head was in Beit El (the source of Rashi is in Bereshit Rabba 69:7). Notice that the southern border of the square
of the oblation passes through Beer Sheba while the northern one passes through
Beit El. Could the ladder of Jacob symbolize the strip of the King-Messiah
around the world? That is why exactly in this place (Gen. 28:14) G-d promised Jacob
"and thou shalt spread abroad to the west, and to the east, and to the
north, and to the south". To the north and to the south- the 12 strips of the tribes, to the
west and to the east- the strip of
Messiah. If our interpretation is correct, the perimeter of the strip of
Messiah around the world should be 8000 parsa, i.e.
32000 mils or 32640 km. Notice a surprising fact. It the strip of Messiah has
perimeter of 32000 mils, its area is 32000x75 mils.
Add to it the calculated area of the 12 tribes (which does not include the area
of the oblation) and we obtain 400x400 + 32000x75=1600x1600 sq. mils, exactly
as claimed by the Talmud.
Now let us calculate the perimeter of the strip of Messiah.
The Temple Mount, more precisely, the center of the
Holy of Hollies lies near the center of the Golden Dome, at the latitude of
31.777719°. The southern border of the oblation is 60 mils or 61.2 km south of
it, at the latitude of 31.2258° while the northern border is 15 mils north of
it, at 31.9157°. The middle of the strip of Messiah is thus 22.5 mils south of
the Temple, on
the latitude of 31.57074°. The cross-section of the Earth on this
latitude is a circle with the radius of 5439.132 km. The perimeter of this
circle is 34175.075 km or 33504.976 mils. It is close to 32000
mils but not very. May be the width of the ladder was calculated based on
π=3? The ladder led to the heaven and there, according to our
understanding, π=3 (in other words, the ladder was a hexagon inscribed in
this circle)! Multiple the above radius
by 6. We obtain perimeter of 31994.89
mils. Not a bad accuracy! It is possible that the ladder surrounded the Earth
not on the sea level but on the level of the mountains of Israel.
Approximately at the latitude of 31.57°, to the north of Hebron lies the highest point of Judea, 1020
m above the sea level (this is exactly one mil, a basic measure in Judaism). On
that level one should add to the radius of our circle
1.02cos 31.5707=0.869 km. The resulting radius will be 5440.001 km and the
perimeter (with π=3) 32,000.006 mil. This is an incredible
accuracy, beyond the definition of the radius of the Earth.
All this is remarkable, but what has it to do with the
ratio ÷å/÷åä (qava/qav)? The strip that surrounds the future oblation of Jerusalem around the
world is akin to a bride that circles around the groom under the canopy. The
spelling of the word curve ÷åä (qava) is
feminine and corresponds to the the bride. The
reading ֌
(qav) is masculine and relates to the groom, the square of Jerusalem. In order to bring these two
measures to a common denominator - the area of Israel, one should multiply the
feminine strip around the world by the ratio ÷åä/÷å (qav/qava),
i.e. 3/π and add to the area of the 12 tribes. If one does not like the
idea to raise the ladder of Jacob to the top of the mountains of Judea, we can suggest a different solution. Out of the
above exact perimeter at the sea level of 33504.98 mils, 75 mils of Jerusalem are
"masculine" and the rest 33429.98 are "feminine". The last
number should be multiplied by the ratio ÷åä/÷å (qav/qava)
equal 106/111 and added to 75. We obtain then 31999.12 mils. A
remarkable accuracy!
The inevitable conclusion is that the one, who designed
this equality, had a full control of the size of the Earth and of the geography
of Israel.