Taken from: http://www.recoveredscience.com/const100solomonpi.htm
The political values of Solomon's wrong Pi
Most current textbooks on the
history of science assert that ancient Near Eastern mathematics were too
primitive for its practitioners to compute an accurate value for the circle
circumference-to-diameter ratio pi. They even claim that the Bible gives the
rather inaccurate value of pi = 3 for this important
mathematical constant.
They refer to the reported
dimensions of the “sea of cast bronze” which king
Solomon placed before the Temple he built in Jerusalem, as described in 1 Kings 7:23:
“It was round in shape, the diameter from rim to rim being ten cubits; it stood five cubits high, and it took a line thirty cubits long to go around it.”
Indeed, the Rabbis who wrote
the Talmud a thousand years after Solomon asserted
this value based on those verses. They may not have been mathematicians, but
they knew how to divide thirty by ten and get three.
Accordingly, they affirmed as late as the middle of the first millennium CE:
“that which in circumference is three hands broad is one hand broad”.
Scholars of the Enlightenment
era were glad to concur with that interpretation because it allowed them to
wield this blatant falsehood in the Bible as an
irresistible battering ram against the until then unassailable inerrancy of the
religious authorities.
Their Colonial-era successors,
in turn, embraced that poor value for Solomon’s pi to belittle the mathematical achievements and abilities of the
ancient non-European civilizations, and to thereby better highlight those of their own modern
Western group. One of the most effective steps in subduing a conquered
nation is to deny or distort its historical achievements, so this poor value of pi in the ancient Near East became rich
fodder for their mockeries.
This parochial attitude received a major blow when the Columbia
University Professor of Comparative Literature Edward
Said published in 1978 his book "Orientalism"
in which he exposed the colonial roots of the then still common Western disdain for the abilities of "Orientals". His
influential comments changed the way some open-minded
literary scholars regarded this biased legacy, but it seems that many
mathematicians and historians of mathematics never got the
memo.
In their domain, the biased
views of those colonialist writers survive to the point that this purported lack
of mathematical intelligence under the reign of a king renowned for his wisdom
is still an article of faith among mainstream
historians of science trained to read this obviously
primitive value into the text.
One of the most popular books
on "A History of Pi" even offers eight translations
of that biblical passage into seven different
languages, presumably to drive home the point with the powerful
mainstream method of proof by repetition, that in
every one of those translations the diameter remains ten cubit and the
circumference thirty1
However, all these disparagers
of Solomon’s pi omit half the evidence. The rest of
the parallel passages they cite from 1 Kings
7:23 and 2 Chronicles
4:2 shows their dogma is based on a hit-and-run calculation of a
type that would make any undergraduates flunk their exam.
It seems that none of those
experts who so compared the diameter and circumference of Solomon’s Sea of
Bronze ever bothered to read on. The next verses, 1 Kings 24 and 26, say that
the circumference was measured under the rim, and that this
rim was flared:
“All round the Sea on the outside under its rim, completely surrounding the thirty cubits of its circumference, were two rows of gourds cast in one piece with the Sea itself. (...) Its rim was made like that of a cup, shaped like the calyx of a lily. When full it held two thousand baths.”
The parallel account in 2
Chronicles 4:3 and 5 leaves out the rim and reads
"Under the Sea, on every side, completely surrounding the thirty cubits of its circumference, were what looked like gourds, two rows of them, cast in one piece with the Sea itself. (...) Its rim was made like that of a cup, shaped like the calyx of a lily. When full it held three thousand baths.”
Obviously, the gourds could not have been under the Sea if they were cast as part
of its circumference on every side, and the measuring rope for the circumference
would not be stretched around the rim where it would
not stay up but only below it. The only practical way to measure such a flared
vessel is to stretch the rope around the body below that
rim.
Moreover, only this measure
directly around the body is relevant for indicating
the volume the vessel could hold, an important part
of its description for which the rim diameter is clearly
irrelevant. It seems therefore that the scribe of 2 Chronicles 4 was
simply as careless in specifying the place of the
measurement as in mis-copying the volume of the
basin.
However, both accounts agree
that the rim was flared. The ten-cubit diameter
measured across its top from rim to rim was therefore
larger than that of the vessel’s body which “took a line thirty cubits
long to go around it”.
The circumference and diameter
reported were thus not for the same circle, and
deducing an ancient pi from these unrelated dimensions would be about as valid
as trying to deduce your birth date from your phone number.
The volume and shape of Solomon's Sea
Moreover, the measuring unit
conversions supplied by modern archaeology allow us to compute the inside volume of that vessel and to thereby
find its shape. With the stated circumference, wall thickness, and height, only
a cylinder can contain the volume of 2,000 bath given
in 1 Kings 7:26.
The cubit
length which had been used in various Jerusalem buildings and tombs of
Solomon’s time was 20.67 inches2, according to the archaeological
architect Leen Ritmeyer who investigated the standards used in those structures.
Like the ancient Egyptian royal cubit of typically similar length, the sacred
cubit used in Jerusalem was also divided into seven hand
breadths of four fingers each.
The bath was a liquid measure of “approximately 22 liters”, as Harper’s Bible Dictionary
states. It was one tenth of a “kor” in the well-known
dry-measuring system which is described in Ezekiel
45:14. Its use for liquids is confirmed by eighth century BCE storage
jars, found at Tell Beit Mirsim and Lachish, that were inscribed “bath” and “royal bath”3. A liter is 61.0237 cubic inch, so 2000 bath equal 304.04 cubic cubit.
As calculated and illustrated
in the above diagram, the 2000 bath of water from
1 Kings 7:26 fill that cylinder close to its top, to
a height of 4.511 cubit above the inside bottom. The outside height was five
cubit, and the bottom was one seventh of a cubit thick, so the 2000 bath leave
only a shallow rim of about 0.3461 cubit above the water
level, or just over seven inches, depending on how accurate the "about 22
liter" conversion factor is.
The rim flare inscribed into the computed rectangle looks
indeed like that of a cup, or like the calyx of a
lily.
|
The height and width of that
rim, computed with the actual value of pi, produce an elegant flare that matches the biblical description. The same holds true for
approximations to pi from about 3 1/8 to 3 1/6 which all produce lily-like rims
and are all closer to the proper value than the alleged but
unsupported pi = three.
These conversions also make it
clear that the copyist of the much later4 parallel history in 2
Chronicles 4:6 misread that volume when he gave it as 3,000 bath. No matter how much you fudge the math or try to
squeeze the incompressible water, this volume does not fit
into a vessel with those dimensions.
Mainstream bias against
non-Western minds
Solomon’s mathematicians and
surveyors, as well as their ancient teachers and colleagues throughout the
ancient Levant, were therefore not necessarily the clumsy
clods portrayed in current history books.
The accuracies of transmitted
lengths which Ritmeyer found in the actual dimensions those ancient builders
left us in stone show that they worked with great
care. It strains credulity that their surveyors could have misread the
rope around that vessel by almost two and a half feet in a circumference of less
than 52 feet.
Nor is there any rational
reason to assume that the ancient number researchers were so innumerate that
they could not have computed a fairly good value of
pi, as close to the real one as that which Archimedes (about 287 to 212 BCE) obtained later, or even
closer. They could wield the same mathematical tool,
the theorem about the squares over the sides of a right-angled triangle
which is now named after the sixth-century-BCE Greek Pythagoras and which
Archimedes used in his pi-calculation many centuries after its real ancient
Babylonian and/or Egyptian authors had discovered it. They also had perhaps
more patience and motivation than Archimedes to continue with the simple but repetitive calculations required for pointlessly
closer approximations.
However, the backwardness of ancient Near Eastern mathematics has become
a cornerstone of the prevailing prejudice against all pre- Greek
accomplishments. Examining that cornerstone exposes the scholarly bias on which it was founded.
The reason for the current denial of ancient pi seems to be that the
calculation of pi requires analytical thinking, the
same exalted mode of thought on which all the rest of so- called Western science
is said to be based, and which must therefore be Western.
Most history books tell us that
this superb achievement and gift to all humanity had to wait for the unique genius of the glorious Greeks, and that the invention of inquisitive and logical thinking was the
decisive contribution from these purported founders of said science.
The Greeks were, in the words
of a highly respected Egyptologist born at the height of the English Empire:
“... a race of men more hungry for knowledge than any people that had till then inhabited the earth”5 .
Reflecting the same then typical attitude which referred to those other people
as “that” instead of “who”, another equally respected historian of science
quoted approvingly Plato’s partisan remark :
“... whatever Greeks acquire from foreigners is finally turned by them into something nobler”6.
The skills displayed in Hezekiah's
tunnel
This cultural bias led some of
the "scholars" afflicted by it not only to disregard obvious facts, as in the
case of Solomon’s pi, but even to fabricate the evidence
they needed to support their supposed superiority. Take, for instance,
the engineering achievement of king
Hezekiah’s tunnel builders.
This biblical king needed to
prepare Jerusalem for a dangerous siege because he
expected a new invasion by the Assyrians who had
conquered the area earlier and extorted from it a heavy tribute which Hezekiah planned to stop paying. To have any chance at all
against this almost irresistible superpower of his
day, he needed to protect the water supply of his city and so had a tunnel dug from inside the walls to the outside
spring.
Because this life-or-death
project was so urgent, the tunnelers started at both ends of that path to then
meet about halfway underground. This unprecedented
mid-way meeting in a more than 1,700-foot-long tunnel would have counted as a
considerable achievement even if their tunnel had
followed a straight line. However, their surveying task was much harder.
At the spring end, the stone
cutters started at an almost right angle to the shortest path towards their goal
and took instead the shortest path towards the city
wall. Maybe they wanted to bring this most vulnerable part of their dig
as quickly as possible under the protection of that wall and of the high
overburden in that area, and maybe they also wanted to take advantage of a few
existing fissures in the rock that happened to run
there for short stretches along their general direction. Then they veered back outside the wall under shallower terrain where
no enemy risked to find the tunnel but where a postulated surface team hammering on the rock above would be easier to hear for
confirmation that the diggers were not straying too far. However helpful and
encouraging these presumed signals from the surface
team may have been to the diggers below, they would have been too diffuse to determine precise locations.
On the town
end of the tunnel, the diggers started northwards but then, instead of continuing north-north-east for the shortest
path towards the other team, they went east and even south-east-east in a wide arc.
Archaeologists call this arc
the "semicircular loop", and some of them suggest
that the diggers took this long detour, adding about 50 percent to the expense
and time for this urgent life-and-death project, to
avoid any possible disturbance to the royal graves of
King David and some of his successors who are said to have been buried in that
area of the "City of David"6A.
The path so prescribed to the
stone cutters became therefore an irregularly curved maximum
challenge to the surveyors who had to multiply their triangulations while
keeping the accumulated errors small enough to not
miss the other team by too far in these two opposite stabs into the three-dimensional dark.
These ancient Hebrew surveyors
solved this complicated task with such skill that we still
don’t know how they did it. Some scholars have argued that they must have
followed a karstic crack underground that went all the way through. However, the
Jerusalem archaeologists Ronny Reich and Eli Shukron
pointed out that the theory of simply following a pre-existing fissure is
incompatible with the several “false” tunnels near the meeting
point7. These indicate
some uncertainty about the path to follow until the teams actually met, and they
are more compatible with the accumulated errors in a small
spread of measuring results.
Moreover, a recent close
examination of the tunnel walls shows that there was
no such continuous fissure. To the contrary, over
long stretches most of the cracks in the rock ran rather at
right or almost right angles to the path of the tunnel8.
That theory about the
continuous crack also ignores the famous inscription
about how elated the cutters were when they at long last heard the voices of the
other group just before they broke through, “axe against
axe”. The joy and relief expressed in that short text would be hard to
explain if the two underground teams had known
beforehand that they were just following a pre-existing path.
Even the authors of the most recent survey of this tunnel, the ones who proposed
the stone cutters might have been guided by the sounds of hammer tapping on the bedrock above the tunnel, do not
think those signals were precise enough to pinpoint the exact locations
underground. One of them admits
"Yet, all things considered, it is quite incredible how the two teams managed to meet almost head-on, at virtually identical elevations as evidenced in the very small difference in ceiling elevation at the meeting point."9
Correctly plotting such a
complicated path underground implies calculating and
measuring skills far better than those attributed to the people whose
predecessors from just two and a half centuries earlier had allegedly misread so
grossly the cord stretched around Solomon’s Sea. It also demonstrates a precision in their trigonometry that does not fit in at all
with their tradition's supposedly so crude pi.
On the other hand, admitting
those skills among Hezekiah’s people would have toppled the
superiority of the Greeks who cut the mostly straight and longer tunnel
of Samos about 170 years later. This tunnel was much easier to measure but
displays much more zig-zagging in the northern leg before the mid- tunnel
meeting10.
So, to prove his contention
that the Israelites worked “in a very primitive way”,
vastly inferior to the “splendid accomplishment” of the Greeks, the above
Plato-buying historian of science invented from whole
cloth a series of vertical shafts he said Hezekiah’s workers had dug from
the top to keep track of their confused and
meandering path11.
This solved
the problem of keeping the Greeks up on their pedestal. Except, of
course, that the veteran Jerusalem archaeologist Amihai
Mazar reports Hezekiah’s tunnel was cut without any such intermediate shafts12. The one and only shaft that is open to
the surface near the southern end of the tunnel is a pre-existing natural feature and not man-made.13
There is still no published
study that explores how Hezekiah’s surveyors could have achieved their stunning success, but the Mathematical
Association of America offers in its 2001 Annual Catalog a video and workbook
about “The Tunnel of Samos” which the Greeks dug less
than two centuries later, also simultaneously from both ends. These discuss the
methods the Greek tunnel builders might have used for “one
of the most remarkable engineering works of ancient times”. And an even
more recent article on this Greek tunnel still relies on the long debunked false assertion about the continuous carstic crack
the ancient Hebrew stone cutters allegedly followed to belittle their even more remarkable achievement:
"The tunnel of Hezekiah required no mathematics at all (it probably followed the route of an underground watercourse)."14
Without this ad-hoc invention of the "carstic crack" and/or "underground
watercourse", the mathematics required for plotting
Hezekiah's tunnel must have been rather impressive.
(In 2011, the above cited
archaeologists Ronnie Reich and Eli Shukron suggested that this tunnel was not
cut during Hezekiah's reign but already under one of his predecessors, perhaps
King Jehoash who reigned from 835 to 801 BCE
14A. However, advancing the date
of this feat by just over a century would not change the argument presented here
that it required mathematics more impressive than what most modern scholars
attribute to Hebrews of that early time.)
Compare those ancient tunnel-builders' skills with
those used in the design and execution of the world's currently longest
mountain-piercing tunnel, through the base of the Gotthard massif in Switzerland, and its final
breakthrough on October 15, 2010. The meeting between the
two opposing sections in this 35.4-mile-long modern tunnel through very hard
rock joined two 31.4-foot diameter bore holes within the specified tolerances of
about 4 horizontal and 2 vertical inches,
matching the ancients' claimed precision of having met "axe
against axe", but in a much longer tunnel.
This spectacular modern
precision was made possible by literally cutting-edge and
space-age technologies developed and/or refined specifically for this
project. This sophistication is only hinted at in the title of its summary
description "The Gotthard Base Tunnel - a challenge for
geodesy and geotechnics"
(http://www.geometh-data.ethz.ch/downloads
/eisenstadt98.pdf).
(http://www.geometh-data.ethz.ch/downloads
/eisenstadt98.pdf).
These surveyors used an unprecedented array of 28 Global Positioning Systems to
nail down the exact locations of their reference points, and the precision of
their specially developed optical surveying
instruments was such that the continuing rise of the Alps by about one millimeter per year became a factor addressed in the
results.
In addition to all their advanced gyroscopic theodolites and opto-electronic X-Y
pickups and other fancy gear, the modern tunnel plotters used also a midway control shaft to confirm
their exact orientation and the elevation of the tunnel
floor despite local variations in gravity, whereas the ancients made do without this help.
Moreover, unlike the ancients,
the moderns credited supernatural help for the
success of their enterprise. Indeed, the inscription that celebrated the
breakthrough meeting of Hezekiah's teams was secular
and mostly technical. It conveys the human excitement of
meeting "axe against axe", but at least the
surviving part of this document from biblical and supposedly pious ancient
Israel contains no mention of God or thanks for
having blessed the project.
By contrast, the celebrations for the Gotthard tunnel
breakthrough in Switzerland, the now reputedly secular birth country of two
major protestant reformations by Zwingli and Calvin,
included the prominent honoring of a statuette representing
the Catholic and Eastern Orthodox Saint Barbara, a traditional protectress from thunder
and lightning. This job associated her later also by default with explosives,
and she thereby became the patroness of miners.
Several of the official tunnel inauguration speakers also
thanked that Saint for her guidance and
supervision of the project.
Their hi-tech measuring
instruments as well as this heavenly help
may have given the modern stone cutters
some advantages over their ancient counterparts so that they could match
the claimed "axe against axe" precision even in
their much longer tunnel.
However, the basic math for plotting their tunnels was identical for
both teams. The moderns may perhaps have applied more powerful
statistical methods for analyzing the scatter of their data points, but their timeless trigonometry and triangulation rules were
still the same as those which Hezekiah's surveyors had clearly used in
their work.
Yet, Orientalism-inspired and
-misled scholars described Hezekiah's surveyors as allegedly innumerate dolts, and they made up imaginary
control shafts, karstic cracks, or buried brooks just to avoid admitting the measuring and computing skills plainly
displayed in the ancients' work. They presume to
judge the richness of the mathematics practiced in the ancient Near East from
only the few surviving and so far deciphered written scraps while excluding the large corpus of unwritten evidence which
demonstrates that the ancients' knowledge was not limited to those few random
tidbits.
Some Western scholars, from
Plato on to this day, needed such fictions to belittle all
pre-Greek achievements and to thereby prove their own and their fellow
Europeans’ superiority over all the other and
allegedly ignorant older civilizations.
The myth of
Solomon’s wrong pi is therefore by now so deeply entrenched in the
Western cultural fabric that most of those who write on this subject continue to
repeat it uncritically because that is what all their
reference books say, no matter how obviously wrong these are.
If you can avoid the blinders
created by this common academic prejudice, you will
see in this book how the allegedly pi-challenged designer(s) of Solomon’s Temple incorporated in the main dimensions of its
layout clear, repeated, and precise evidence that
their pi was at least as good as that of Archimedes.
Moreover, their teachers, as
well as the ancient Egyptian inventors of their
mathematical methods, had also computed several other important mathematical
constants with remarkable precision. And they had invested these transcendental
numbers with transcendental meanings that
revealed to them the inner workings of their world.
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