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.
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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 BCE14A. 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).
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.
