The Value of Pi and the Circumference of the "Molten Sea" in 3 Kingdoms 7,10

One of the appurtenances fabricated for King Solomon"s temple complex by Hiram of Tyre was the "molten sea" – an immense water basin of cast bronze. The dimensions of the huge vessel are given in 1 Kgs 7,23.26:

And he made the molten sea, ten cubits from brim
to brim, round in compass, five cubits high, and a line
of thirty cubits measured its circumference. . . .
Its thickness was a handbreadth; and its brim was like the
brim of a cup or the blossom of a lily . . .

The cubit (hm)) represents the length of a straightened forearm from the elbow to one of the fingertips; the handbreadth (xp+) is one-sixth of a cubit ¹.Dividing the 30 cubit circumference by the 10 cubit diameter gives 3, a value for pi that had long been used by the Babylonians ².

The corresponding LXX passages are 3 Kingdoms 7,10.12. Although the diameter, height, and wall thickness of the sea correspond to the dimensions given in the MT, the circumference does not; it is given in 3 Kingdoms 7,10 as "thirty-three cubits" instead of "thirty cubits". Why the discrepancy? Although this is a very provocative question, it has largely been neglected by OT scholars.

One exception is R. B. Y. Scott, who interprets the measurements as belonging to two different circles. Taking the 10 cubit diameter as an inside measurement, Scott surmises that the outside diameter of the sea must have been 10 1/3 cubits because the walls were one handbreadth thick, and a handbreadth is one-sixth of a cubit: 1/6 + 10 + 1/6 = 10 1/3. Multiplying this 10 1/3 cubit outside diameter by a pi value of 3 1/7 gives an outside circumference of 32.47 cubits or 33 cubits when rounded to the next greatest whole number ³.

The great Greek scientist Archimedes (287-212 BC) had determined that the value of pi lies somewhere between 3 10/71 and 3 1/7 ⁴. The upper limit of 3 1/7 became, and still remains, a convenient approximation for pi itself. Archimedes had spent time in Alexandria – where the LXX translations were being undertaken – and kept up correspondence with other Greek intellectuals there. His 3 1/7 approximation of pi would have been considered the state of the art value in Alexandrian mathematical circles. It is not at all unreasonable to assume that well-educated, Greek-speaking, Alexandrian Jews – such as a LXX translator or editor – could have become familiar with the value as well.

Although Scott"s calculation of a 32.47 cubit circumference from the assumed diameter and pi value is correct, his rounding of that value up to 33 cubits is simply not tenable. The 32.47 cubit figure can for all practical purposes be considered as 32 1/2 cubits or "thirty-two cubits and a span". The half-cubit span (trz, Greek spiqamh\) was the distance between the tip of the thumb and the tip of the little finger of a spread hand. One of the most well-known measurements in the Bible, the height of Goliath, is given in the format "so many cubits and a span" (1 Sam 17,4). It is unlikely that the LXX translator/editor – after meticulously factoring wall thickness into his diameter calculation, multiplying that diameter by a state of the art pi value and coming up with an answer that is an almost perfect "so many cubits and a span" figure – would have then compromised such an accurate result by needlessly rounding up to the nearest whole cubit.

Furthermore, the premises on which Scott predicates his assumed diameter may not be tenable either. According to the given description, the brim of the sea was like the brim of a cup or the blossom of a lily. In other words, the walls of the sea flared out around the top of the sea to form the brim. The maximum outside diameter of the vessel as measured from brim to brim would therefore have to be greater than Scott"s 10 1/3 cubit measurement which takes into consideration only the thickness of the walls but not the width of the brim itself.

The maximum outside brim to brim diameter can easily be calculated by dividing the 33 cubit circumference by the 3 1/7 pi value. The result is exactly 10 1/2 cubits, a perfect "ten cubits and a span." This cannot have been mere coincidence; on the contrary, it demonstrates that the LXX translator/editor deliberately chose a number that when divided by 3 1/7 would yield the precise figure of 10 1/2. A 10 1/2 cubit outside diameter indicates that the width of the brim encircling the top of the sea must have been exactly one-fourth of a cubit: 1/4 + 10 + 1/4 = 10 1/2.

It should be noted that the LXX translator/editor did not have to tamper with the circumferential measurement in order to make the passage conform to the more accurate pi value of 3 1/7 instead of the less accurate value of 3. By interpreting the given 10 cubit diameter and 30 cubit circumference as two separate measurements and then dividing the circumference by 3 1/7, he would have come up with an inside diameter of approximately 9.54 cubits, indicating a brim width of approximately 0.23 cubit. This is essentially the same interpretation of the measurements later given by Rabbi Nehemiah in the Mishnat ha-Middot, a Hebrew geometry text dating from c. AD 150 ⁵. The reason for the LXX translator/editor"s not accepting the 30 cubit inside circumference at face value may lie in the text"s statement that a line (hwq, "measuring line") of 30 cubits measured the circumference of the sea. A measuring line can easily be stretched around the outside of a round vessel, but not the inside of a round vessel. Sufficiently increasing the length of the 30 cubit circumference converts it from an inside measurement to an outside measurement, thereby making it possible to stretch a measuring line around it. After a few trial divisions by 3 1/7 of tentative circumference measurements greater than 30 cubits, the LXX translator/editor would have hit upon this "perfect" measurement of 33 cubits. He furthermore could have rationalized his actions by supposing that he was emending a corrupt text: Perhaps a careless copyist, intending to transcribe an original circumference of "thirty and three" cubits, had omitted the "and three" and had simply written "thirty" ⁶.

A possible objection could be raised against this analysis on the grounds that there are significant differences between the material in Kings and Kingdoms, including other discrepancies in measurement. In fact, one of these discrepancies involves another circumferential measurement – that of the large twin columns flanking the temple entrance. 1 Kgs 7,15 gives the measurement as 12 cubits, but 3 Kingdoms 7,3 gives it as 14 cubits. The implication is that the LXX translator/editor could have been working from a Vorlage which perhaps did give the circumference of the sea as 33 cubits. The 33 cubit circumferential measurement is more than just a variant of a particular dimension, however. Unlike the circumferential measurements of the columns, the circumferential measurement of the sea is accompanied by a given diameter – and whenever a given diameter appears in conjunction with a given circumference, there is a definite possibility that an implicit statement about pi is being made.

If the author of the Vorlage intended the measurements to describe the same circle (e.g., maximum outside dimensions), he would be implying a pi value of 33 divided by 10 = 3.3, a value even less accurate than the old Babylonian value of 3 and a value nowhere else attested in antiquity ⁷. If, however, he intended the measurements to describe the two separate circles comprising inside and outside dimensions, the measurements would still reflect the old Babylonian value of 3: Dividing the 33 cubit circumference by 3 gives an outside diameter of 11 cubits, indicating a brim width of exactly one-half cubit or one span. Then again, there is no reason why such a Vorlage could not have given the same 30 cubit circumference as the MT. After all, this is the figure that repeatedly shows up in other sources such as Chronicles, the Mishnat ha-Middot, and the Talmud; the 33 cubit figure never shows up again after its appearance in 3 Kingdoms. A 30 cubit figure in the Vorlage would mean that all the arguments for the LXX translator/editor"s emendation of the text to read 33 cubits and reflect a 3 1/7 pi value would still apply. Given the common Alexandrian background of both the LXX and the 3 1/7 value for pi, this latter view would seem to be the most plausible.

Summary

The dimensions of the "molten sea", the huge vessel fabricated for King Solomon"s temple, are given in 1 Kgs 7,23.26 (MT) and 3 Kingdoms 7,10.12 (LXX). All measurements of the MT correspond exactly to those of the LXX except one, the circumference. The MT gives "thirty cubits" and the LXX "thirty-three cubits". It seems probable that the MT used the value attributed to pi by the Old Babylonian (pi = 3), whereas the LXX may have known the more accurate value discovered by Archimedes and presumably known in Alexandria (pi = approximately 3 1/7).

¹ M. A. POWELL, "Weights and Measures", ABD, 6 (New York 1992) 899-900. The Greek cubit (ph/xuj) and its subdivisions proportionally correspond to their Hebrew counterparts.

² L. N. H. BUNT – P. S. JONES – J. D. BEDIENT, The Historical Roots of Elementary Mathematics (Englewood Cliffs 1976; reprint New York 1988) 61-62.

³ R. B. Y. SCOTT, "The Hebrew Cubit", JBL 77 (1958) 209-210.

⁴ BUNT et al., Historical Roots, 196.

⁵ For text, translation, and commentary, see S. GANDZ, Studies in Hebrew Astronomy and Mathematics (New York 1970) 349.

⁶ Modern exegetes have also attempted to rationalize the measuring line stretched around the 30 cubit circumference: A. E. BERRIMAN, Historical Metrology (London 1953) 97, points out that large castings of this type were made upside down, with the core of the mold forming the inside of the vessel. Stretching a measuring line around the outside of this core would give a measurement that would correspond to the inside circumference of the cast vessel. A. ZUIDHOF, "King Solomon"s Molten Sea and (p)", BA 45 (1982) 179-184 maintains that both the diameter and the circumference were exterior measurements of a cylindrical sea with a flared-out brim. Because of the difficulty involved in stretching a measuring line around the narrow brim, the measurers instead stretched the line straight across the top of the sea to get a maximum outside diameter from brim to brim. They then took a circumferential measurement around the outer vertical walls of the sea somewhere below the brim.

⁷ This value is actually cited by J. A. MONTGOMERY, The Books of Kings (ICC; Edinburgh 1951) 153.