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Biblical measuring units and Solomon's "Sea"
by Prof. Mordochai ben-Tziyyon, Universitah Ha'ivrit, Y'rushalayim

BIBLICAL MEASURING UNITS

Units of length

Three units of length are commonly found in the Scriptures, namely the טֶפַח tefah (translated as "handbreadth" by christians), the זֶרֶת zeret ("span") and the אַמָּה ammah ("cubit"). One tefah is the width of the four finger's of an average-sized man's hand, and is about 83mm (a little over three inches). A zeret is 3 tefahs, i.e. about 25cm (approx. 9¾ inches), and an ammah is 2 zerets or 6 tefahs, i.e. about 50cm (approx. 19½ inches). The Talmud refers to the distance of 2,000 ammahs (specified in B'midbar 35:5 as the width of open land surrounding a city that is deemed to be part of the city) as a מִיל mil or "Talmudic mile" and is equal to 1.0km (0.621 mile or 1,093.6 yards).

Shmuel Alef 17:4 records that Golyat of Gat (the giant that David killed using only some kind of catapult or sling and one small stone) was "six ammahs and one zeret tall" - that is to say, 6½ ammahs, or about 325cm (approximately 10'8"). Now that's a BIG man!!!

Units of weight

The principal units of weight found in the Scriptures are the שֶׁקֶל shekel and the כִּכָּר kikkar (translated as "talent" by christians). The relationship between these two units may readily be determined from Shmot 38:25-26, where it is recorded that the first census yielded a total of 100 kikkars and 1,775 shekels, corresponding to a contribution of one בֶּקַע beka, or half a shekel, from each of the 603,550 adult males in the nation's population. Now 603,550 half-shekels equal 301,775 shekels, so it follows that 300,000 shekels are equal to 100 kikkars - i.e. one kikkar is equal to 3,000 shekels.

Another unit of weight that was used in Talmudic times was the maneh, for which there were two standards - the "Italian maneh", equal to one-sixtieth of one kikkar, and the "temple maneh", of which 37½ were equal to one kikkar. The latter was used to measure the 11 ingredients of the k'toret (see Babylonian Talmud, treatise K'ritut, folio 6a and Y'rushalmi Talmud, treatise Yoma, chapter 4, halachah 5) and was equal to 80 shekels, or a little over one kilogramme (about 20 ounces).

It is recorded in M'lachim Alef 10:14 that King Shlomoh's annual taxation revenue was 666 kikkars, i.e. 666×3,000 = 1,998,000 shekels, of gold. The weight of a shekel is known with considerable reliability because numerous specimens of silver shekel coins from the second Temple period, and also from the brief three-year reign of Shim'on ben Kuziva, also known as "bar Kochba" (132-135BCE) have been unearthed by archæologists - a shekel weighed approximately seven grammes (one-quarter of an ounce); thus, King Shlomoh's annual taxation revenue was equal to about half a million ounces, or roughly 14 tonnes, of gold - a truly colossal sum!

Another passage in which kikkars are mentioned occurs in Divrei ha-Yamim Beit 25, where it is recorded that in preparation for his attack on the Edomites in which he captured Petra (compare M'lachim Beit 14:7), King Amatzyahu of Y'hudah (reigned 824-796BCE, but in name only after 810BCE) had hired 100,000 mercenaries from the northern kingdom (Divrei ha-Yamim Beit 25:6) to augment his own forces of 300,000 warriors (v.5), but had dismissed them (v.10) after being warned by an un-named prophet that he would be defeated if he allowed them to fight alongside the Y'hudæans (verses 7-8). Amatzyahu had paid the northern mercenaries 100 kikkars of silver for their services (v.6) and, in an amusing human note, he asks the un-named prophet, "So what about my one hundred kikkars of silver?", to which he receives the terse response: "Adonai can give you much more than that!" (verse 9). Since one kikkar is equal to 3,000 shekels, it follows that Amatzyahu's mercenaries had cost him 300,000 shekels (i.e. three shekels for each of the 100,000 mercenaries) - more than two tonnes - of silver, so it's hardly surprising that he was concerned about getting it back!

It is difficult to gauge in terms of modern currency what the monetary value of a silver shekel (i.e. one shekel weight of silver) would have been in Biblical times. Although the economies of ancient Yisrael and Y'hudah do not appear to have been subject to inflation in the same way that modern economies are, they were certainly affected by market forces of supply and demand: for example, during the Aramean siege of the northern Hebrew capital Shomron ("Samaria") in the time of King Y'horam ben Ah'av (reigned 881-870BCE), there were such severe shortages in the city that the price of a donkey's head rose to 80 shekels and the price of one-quarter kav (just under 400ml or about 13 fluid ounces) of pigeon-droppings - which the people were forced to use as fuel as they could not leave the city to get wood because of the siege - rose to five shekels (M'lachim Beit 6:25), and after the siege was broken the prices of grain fell back to their normal levels - one shekel per sé'ah (approximately 10 litres or a little over two gallons) for wheat-flour and half a shekel per sé'ah for barley-meal (M'lachim Beit 7:16).

Historical digression

From this it emerges that a shekel of silver was a relatively small sum of money, and it seems surprising therefore that Amatzyahu had been able to hire those 100,000 mercenaries for only three shekels each. This tells us something about the economic conditions in the northern kingdom at that time; unemployment and poverty must have been very severe for so many men to have been willing to sign up to risk their lives fighting for the neighbouring king in return for such a paltry sum. To make matters worse, Amatzyahu had then summarily dismissed the northern mercenaries before they ever got near a battle-field (although it's clear that they had already been paid their three shekels) and, enraged by this added insult, the northern mercenaries had gone on a rampage of killing and plundering through the cities of northern Y'hudah (Divrei ha-Yamim Beit 25:13). Because Y'hoash, the northern king (reigned 825-812BCE), did nothing to prevent this, or even to punish those responsible, Amatzyahu declared war on him (M'lachim Beit 14:8, Divrei ha-Yamim Beit 25:17).

Y'hoash, who as we have already noted had enough domestic problems of his own, had no desire to fight his brothers in the south but Amatzyahu marched his troops towards Shomron, so Y'hoash was forced to invade Y'hudah to confront the approaching Y'hudæan forces and the two armies met at Beit Shemesh in northern Y'hudah (M'lachim Beit 14:11, Divrei ha-Yamim Beit 25:21). Amatzyahu was defeated and his soldiers deserted him (M'lachim Beit 14:12, Divrei ha-Yamim Beit 25:22) and he himself was captured, after which Y'hoash marched on Y'rushalayim and plundered the Temple and the Royal Treasuries (M'lachim Beit 14:13-14, Divrei ha-Yamim Beit 25:23-24). Y'hoash died soon after that and Amatzyahu either escaped or was released; he outlived Y'hoash by 15 years (M'lachim Beit 14:17, Divrei ha-Yamim Beit 25:25). But after his defeat and capture at Beit Shemesh, Amatzyahu seems to have lost the confidence of his subjects. M'lachim Beit 14:17-21 records that

and the version in Divrei ha-Yamim Beit 25:25-26:1 has a few extra words...

suggesting that the uprising against Amatzyahu, his flight to Lachish, and the popular crowning of his son Azaryah (who is called Uzziyahu in Divrei ha-Yamim) in his place, all coincided with Y'hoash's death (which occurred in Amatzyahu's 15th year), and that he spent the whole of the last 15 years of his reign (that is, from his 15th until his 29th inclusive) in hiding at Lachish while his son Azaryah/Uzziyahu ruled as regent.

Units of volume/capacity

The following units of capacity are found in the Bible:

• רוֹבַע rova or לוֹג log‡,
• קַב kav,
• עוֹמֶר 'omer or עִשָּׂרוֹן 'issaron‡,
• הִין hin,
• סֵאָה sé'ah,
• אֵיפָה eifah or בַּת bat‡, and
• חוֹמֶר homer or כּוֹר kor‡.

‡ Some of these units have two names because one was used for dry measure and the other for liquid measure - for example, one might have measured an eifah of flour and a bat of water, but both would have had the same volume (see Y'hezkel 45:11).

one rova (dry measure) or log (liquid measure) was equal to the combined volume of six average-sized chicken eggs

one kav was equal to four rovas (dry measure) or logs (liquid measure)

one hin was equal to three kavs

one sé'ah was equal to two hins

one eifah (dry measure) or bat (liquid measure) was equal to three sé'ahs

one homer (dry measure) or kor (liquid measure) was equal to ten eifahs or bats

one 'omer (dry measure) or issaron (liquid measure) was equal to one-tenth of an eifah

An egg's volume can be determined by measuring the amount of water displaced when it is completely submerged in water, or more accurately by submerging a large number of eggs at the same time, measuring the quantity of water displaced, and then dividing by the number of eggs used - it is found to be about 65 cubic centimetres (or equivalently 65 millilitres) - we therefore have

Relationship between units of length and units of volume/capacity

The Talmud states that the smallest mikvah (ritual immersion tank) in which it is possible for an average-sized man to immerse himself totally needs to be one ammah by one ammah by three ammahs deep, equal to a capacity of forty sé'ahs (Treatise Eiruvin folio 4b and numerous other places). It therefore follows that 3 cubic ammahs are equal to 40 sé'ahs, which is correct because

3 cubic ammahs ≈ 3×50×50×50 cm³

3 cubic ammahs ≈ 375,000 cm³

3 cubic ammahs ≈ 375,000 ml

3 cubic ammahs ≈ 375 litres,

and 40 sé'ahs ≈ 40×9.36 litres = 374.4 litres.

 
SOLOMON'S "SEA"

Solomon's yam ("Sea") was a huge copper tank that was used by the kohanim for performing ritual immersions. Two parallel descriptions of it are given, one in M'lachim---

and one in the Divrei ha-Yamim ---

These two parallel descriptions of the "Sea" do not specify what material it was made of, but we know from M'lachim Beit 25:13, Yirm'yahu 52:17, Divrei ha-Yamim Alef 18:8 and other references that it was made of copper. The two passages are almost identical, apart from the terms used for the two rows of ornamental decorations; these are called פְּקָעִים p'ka'im or "buds" (KJV "knops") in M'lachim and דְּמוּת בְּקָרִים d'mut b'karim or "figures of cattle" in the Divrei ha-Yamim . There is one other apparent discrepancy, namely the capacity of the "Sea", and I shall talk about that later.

Solomon's "Sea" is often depicted pictorially as a round-bottomed hemispherical tub, but it cannot have been because a half-sphere with a diameter of 10 ammahs and 5 ammahs deep (at the centre) would have held less than 1,200 bats of water, and it is stated explicitly that the "Sea" held 2,000 bats. Moreover, the two rows of ornamental "buds" extended across the entire 10 ammahs of each side "below the brim", so clearly the lower part of it was square and only the top portion was circular. The Talmud records that the lower three ammahs of the total depth of five ammahs were square, and the upper two ammahs were circular, like this:

It has been pointed out many times that M'lachim Alef 7:23, as it appears in "King James's Per-Version", is mathematically impossible. According to that translation, the "Sea" was...

which implies that π = 3. The problem with King James's translation is that the verse actually says וְקָו שְׁלֹשִׁים בָּאַמָּה יָסֹב אֹתוֹ סָבִיב v'kav sh'loshim ba'ammah yasov oto saviv, i.e. "a line 30 ammahs [long] went all the way round it". The KJV wording "did compass it round about" gives the impression that the exterior circumference is meant, but the Hebrew words convey no such sense: they could refer equally to the interior or to the exterior circumference. Now verse 26 states that the copper walls of the "Sea" were one tefah (one-sixth of an ammah) thick, so the interior radius was only four and five-sixths ammahs. The interior circumference was therefore

or thirty ammahs to the nearest ammah, as stated in both M'lachim and Divrei ha-Yamim.

Ignoring the thickness of the copper walls, the cubic capacity of the "Sea" is found to be

(10² × 3) + (π × 5² × 2)  cubic ammahs

=  300 + 50 × 3.14159  cubic ammahs

=  300 + 157.0796  cubic ammahs

=  457.0796  cubic ammahs

=  152.3598  mikvahs [because one "mikvah" is equal to three cubic ammahs]

=  6,094.392  sé'ahs [because one "mikvah" is also equal to forty sé'ahs]

=  2,031.464  bats [because one bat is equal to three sé'ahs]

which is very close to the value of 2,000 bats given in M'lachim Alef 7:26. However, if the the thickness of the copper walls is taken into account, the actual capacity of the "Sea" is only 411.5410 cubic ammahs = 137.1803 mikvahs = 5487.212 sé'ahs = 1829.071 bats.

Finally, I promised earlier to talk about the discrepancy between the capacity of the "Sea" given in M'lachim and the value given in Divrei ha-Yamim . Apart from the different numbers, there is a significant difference in wording between M'lachim Alef 7:26 and Divrei ha-Yamim Beit 4:5 - the former merely says אַלְפַּיִם בַּת יָכִיל alpayim bat yachil, literally "it could contain 2,000 bats", while the latter says מַחֲזִיק בַּתִּים שְׁלֹשֶׁת אֲלָפִים יָכִיל mahazik batim sh'loshet alafim yachil, literally "it held bats, it could contain 3,000 [measures]". This somewhat cryptic statement tells us two things: first, that the "Sea" could hold a large quantity of water (because bats are liquid units), but second, that it could hold three thousand of some unspecified units.

It turns out that the "unspecified units" are eifahs, the dry measure equivalent of liquid bats. In other words, although the "Sea" was designed to contain liquid, and could hold 2,000 bats of water, it was capable of holding 3,000 eifahs of dry material, heaped (from which the Talmud derives an empirical rule that the overflow in a heaped measure is equal to one-third of the total amount).

Postscript: the value of π

Even though, as I have already demonstrated, there is no discrepancy between the circumference of the "Sea" given in M'lachim and the Divrei ha-Yamim and mathematical theory, the text nevertheless does seem to imply that the value of π is exactly 3, which we know is not so. However, hidden in the Masoretic tradition is a possible hint at a much better approximation, because M'lachim Alef 7:23 contains one of the many scribal spelling anomalies that occur in the Scriptural text where the k'rei (how a word is meant to be pronounced) differs from the k'tiv (the way it is actually written). Here is how M'lachim Alef 7:23 appears in a scroll of M'lachim written by a sofer for use in ceremonial public readings, with the words v'kav sh'loshim ba'ammah ("and a line of thirty ammahs") highlighted:

The Talmudic word גֵּימַטְרִיָּא geimat'riyya comes from the Greek word γεωμετρια geometria, a combination of γη ge (the Earth) and μετρω metro (to measure), and originally meant "measurement of the Earth"; although at a very early stage the term acquired a much wider application and came to refer to the study of the properties of all points, lines, surfaces and solids in space and the relationships between them. But geimat'riyya has nothing to do with geometry; it's a Rabbinic exegetical (i.e. interpretive) device in which the letters of a Hebrew word or phrase are treated as numbers: the first nine letters of the alphabet represent the numbers 1 through 9, the next nine represent 10 through 90 in steps of 10, and the last four represent 100 through 400 in steps of 100 - as follows:

What does this have to do with Solomon's "Sea"? Well, the word kav, a line, is normally spelt קו or kuf (100) vav (6), but in M'lachim Alef 7:23 the Masoretic spelling is קוה or kuf (100) vav (6) hé (5). In other words, in the phrase "a line of thirty ammahs", the geimat'riyya "value" of the word line is artificially inflated from 106 to 111 - and if we take the hint and increase the stated figure of thirty ammahs in the ratio 111:106, we get

which differs from the mathematically "precise" circumference (31.416 ammahs) by only one-thousandth of an ammah (0.5mm, or less than one-fiftieth of an inch). This suggests an approximation to π of 3.14151, considerably more accurate than the famous result due to Archimedes (3rd century BCE), namely

which only estimates π as lying somewhere between 3.1408 and 3.1429.

See also: Solomon's kiyyors ("basins")