King Solomon and the value of Pi

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15 Shevat 5772 / February 8, 2012

Did King Solomon not know maths?

King Solomon is said to have been the wisest man that ever lived. Yet critics of the Bible gleefully point to a seemingly glaring error which apparently indicates that he did not know the true value of π (Pi), which as every schoolchild knows is 3.14..

Pi stands for the Periphery (or perimeter, circumference) of a circle, which is in the ratio of approximately 3.14 to its diameter regardless of the size of the circle. This is one of the most important mathematical constants and is used in many mathematical, scientific and engineering formulae.

π is called an “irrational” number, which means that its value cannot be expressed exactly as a fraction having integers in both the numerator and denominator (unlike 22/7). Consequently its decimal is never-ending and never repeats.

Ancient papyri from Egypt and tablets from Babylon are said to prove that 4000 years ago the value of pi was known to be approximately 256/81=3.16 or 25/8=3.125, which are within 1 percent of the true value. However, many historians of mathematics consider Archimedes (287-212 B.C.E.) to have been the first to estimate pi rigorously at about 3.14185. With modern computers the value of pi is said to have already been calculated to 5 trillion decimal digits.

On the other hand, the Bible, as we shall presently see, makes it appear as if King Solomon was stumbling hopelessly far behind those who came before and after him.

According to the rabbinic dating system of the ancient midrash, Seder Olam, King Solomon was born in the Hebrew year 2912 (-848 B.C.E.), and came to the throne at the tender age of twelve in 2924 (-836 B.C.E.). It was four years later in the year 2928 (-832 B.C.E.) when at the age of 16 he began to build the Temple in Jerusalem.

So it may not seem too surprising if this young lad was way off in the accuracy of his estimate of pi, even though the Egyptians and Babylonians had apparently known far better over a thousand years earlier. Compared to the superior Greek Archimedes, who lived 600 years later, it looks as though King Solomon was just fumbling in the dark.

What was King Solomon’s estimate of the Diameter-Circumference ratio?

The New International English translation of the Bible in I Kings 7:23 says:

23 He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.

This “Sea” was the “Pool of Solomon”, an immense bronze circular pool of water supported on twelve bronze oxen. The priests would immerse in the water to purify themselves before beginning their services in the Temple (see II Chronicles 4:6). If the diameter of the pool was ten cubits (appx. 5 meters), saying that it took a “line of thirty cubits to measure around it” seems like a very crude approximation of the value of pi as a round 3. Without knowing that pi is really 3.14.. and that the circumference of the pool needed to be 31.4.. cubits, how could Solomon’s master bronze worker Hiram make a perfectly circular pool?

Related questions are indeed addressed by the Talmudic sages, whose deliberations on the value of pi appear in Tractate Bavli Eiruvin 13b and are also involved in the discussion on the area of a circular Succah in Bavli Succah 8a.

The Talmudic sages appear to have had the same woefully approximate estimate of pi, as in the words of the mishnah in Eruvin 1:4: “Every circular object that has a circumference of three hand-breadths has a diameter of one hand-breadth.” The sages traced this principle to the cited verse from I Kings 7:23 (see Eiruvin 14a), and even they understood how much of an approximation it is, asking if the “line” used to measure round the Pool was inside or outside its decorative bronze lip.

Woe to the blind and ignorant who jump to false conclusions based on half-digested readings of Bible translations, without being able to understand the profound subtlety of the original Hebrew text.

I Kings 7:23

A careful examination of the Hebrew text of the key verse in I Kings 7:23 indicates that over 2800 years ago, King Solomon embedded in the written word of the prophet the value of pi as 3.14150...which is accurate to four decimal places (modern calculations put the value of pi at 3.14159..). Solomon’s calculation is way superior to the estimates of 3.12 or 3.16 by the Babylonians and Egyptians who preceeded him, and compares very favorably with one of the earliest accurate calculations of pi as 3.1416 by the Chinese mathematician Liu Hui around the year 265 C.E.

The embedding of this mathematical truth in the Bible is an open secret that is perfectly visble in every Hebrew Bible but comprehensible only to those who are familiar with the Hebrew letters and their mathematical values (gematria).

In the key verse in I Kings 7:23, the Hebrew word for the “line” that went around King Solomon’s Pool is pronounced as KaV, as indicated in the redded Kri, in the left hand margin of the attached image of the relevant passage scanned from the authoritative Koren Hebrew Bible. This reading of the word is the Mikra, (“reading” or “declamation”) as opposed to the Masora, the scribal tradition as to how the word is to be spelled in the actual text of the book of Kings when written on parchment. There are numerous places in the original. Hebrew texts of the Five Books of Moses, Prophets and Holy Writings (TaNaCh) where tradition requires that the word be pronounced one way (Kri) while the Ktiv (“writ”, the actual script) is written differently.

In this verse as seen in the image, the Ktiv can be seen reddened in its proper place in the Hebrew text exactly as it is would be written by the sopher (scribe) on a parchment scroll of the book of Kings. (The klaf or “scroll” does not include the vowel and cantillation marks found in the Koren and other Hebrew printed Bibles.) Besides the normal Hebrew letters of the word KaV, Koph-Vav, the Masora prescribes the writing of an additional Heh at the end of the word, making it K_V_H, although it would not be pronouced that way in the declamation but as KaV in accordance with the tradition of the Mikra.

When the reader would declaim the text according to the kri, his audience – many of them simple people – would know that the approximate ratio of the diameter of a circle to its circumference is 3. There is really no need to bother ordinary lay men, women and children with a string of 5 or more decimal places.

But educated scholars who venerate every word and letter of the holy writ would know from the fact that the ktiv is different from the kri that something is afoot.

The gematria (sum of the numerical value of the Hebrew letters) of KaV is koph-100 + vav-6 = 106.

The gematria of KaVaH is koph-100 + vav-6 + Heh-5 = 111.

Expanding the value of the line or Kav in the ratio of 3 x 111/106 gives us the value of pi as 3.14150. Compared with Archimedes’ estimate of 3.14185 made 600 years later, King Solomon was well in advance!

He Who said to His Universe: “Enough!” 

The Midrash states that when God created the Universe, it continued expanding endlessly until He said “Enough!” (Bereishit Rabba 46:3; see Rashi on Genesis 43:14). Perhaps we may think of lines of light wanting to extend endlessly straight ahead in all directions, but a divine power would not allow this and constrained them to veer sufficiently from their straight path to bring them back eventually in a circle to their Source, even though it may appear to tiny little space- and time-bound creatures like ourselves that the lines are straight.

To prevent an extending line going straight forward forever but instead to constrain it to veer exactly enough to go in a great circle that will eventually return to its beginning, it is necessary to apply a force involving the ratio of pi.

In the Hebrew Torah, the aspect of God that wields this constraining force is called by the name of ShaDaI, which as taught by the Torah sages alludes to the fact that He is the One She-amar Le-Olamo Dai , “Who said to His universe: Enough”.

Is it mere coincidence that the Gematria of ShaDaY (Shin-300 + Dalet-4 + Yod-10) is 314?

Rabbi Nachman and the Maskilim of Uman

As a footnote to this discussion, lovers of Rabbi Nachman may find it interesting that the mathematical problems concerning the circumference and area of circles discussed in the Talmud in Eiruvin 13bff and Succah 8a were the subject of Rabbi Nachman’s Shabbat table discourse when he was first passing through the town of Uman, Ukraine in the year 1802 when he was aged 30. It was during that visit that he set his heart on being buried there in the old Jewish cemetery amidst the graves of the martyrs of the Uman massacre of 1768. Although Rabbi Nachman was to spend the next 8 years in Breslov, he eventually moved to Uman, where he died in 1810, and his grave is there until today.

In 1802 the Jewish community of Uman was under the sway of a clique of Jewish atheists who had succeeded in preventing the Chassidic leader Rabbi Levy Yitzchak of Berdichev settling in the town as Rav.

During Rabbi Nachman’s Shabbat stay, the atheists wanted to check out this young great grandson of the Baal Shem Tov, and attended his table. As the Rebbe directed his discussion to Talmudic mathematics, the atheists were amazed that a Chassidic rabbi could have such a grasp of the intricacies involved. They insisted on meeting with him again, and became so enchanted that they became his friends and are said eventually to have repented. See Until the Mashiach by Rabbi Aryeh Kaplan (Breslov Research Institute) pp. 73-76.

About the author

Avraham ben Yaakov

Jerusalem-based Avraham ben Yaakov is a Torah teacher & author of translations and commentaries on Bible, Hassidut, Kabbalah, Spiritual Growth, Health & Healing.

By Avraham ben Yaakov

Avraham Ben Yaakov

Jerusalem-based Avraham ben Yaakov is a Torah teacher & author of translations and commentaries on Bible, Hassidut, Kabbalah, Spiritual Growth, Health, Healing and the Environment.

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