Language Studies

Difficult Sayings

Bad Maths in the Bible: the value of Pi
1 Kings 7:23

"And he made the Sea of cast bronze, ten cubits from one brim to the other; it was completely round. Its height was five cubits, and a line of thirty cubits measured its circumference." (1 Kings 7:23, KJV)

Every schoolboy or girl knows of פ Pi, very few can quote it beyond a few decimal places, although several have mastered Pi to between ten and forty thousand digits as record breaking attempts (http://pi-world-ranking-list.com/lists/memo/index.html). What most teenage children or adults with at least some recollection of geometry will tell you, though, is this, namely that Pi is an irrational number somewhat bigger than the apparent value of 3 given here. The age old calculation of circumference over diameter equals Pi, here gives us 30/10=3.0, whereas Pi is 3.1415926 and a whole host more digits (more than 206 billion known at the current count, http://www.lupi.ch/PiSites/Pi-Rekord.html).

In Abraham's day the Egyptians had approximated Pi to (4/3)^4 = 3.16 (to 2 decimal places) and the Babylonians used the value 3 1/8 (=3.125). In India they were using the square root of 10 = 3.162. So the Bible's estimate is out by a fair margin even by ancient mathematical standards. This has led to many sceptics quoting this verse as evidence against the Bible's trustworthiness. Only a few weeks ago in the London Daily Telegraph (25 April 2004) a correspondent wrote a letter "No פ in the sky" refuting the dimensions of Solomon's Temple and arguing against the inspiration of the Bible on the basis of the geometrical error.

The first defensive explanation often given is that the Bible uses rounded numbers to represent fractions. Besides that, the Hebrew unit of measurement known as the cubit was itself notoriously vague and given as the length of a man's forearm from his elbow to his fingertips. This ranged from 12 to 18 inches depending on the man! Thus, we are unable to say for certain how big Noah's ark was, although it was huge and was the largest ship known to man until the twentieth century. If an ancient Hebrew were building a house and had to change builders mid-project he would need to get another builder with similar length arms just to keep the measurements consistent! Applied to our dilemma above 31.4 cubits may well have approximated to 31 or 30 cubits. Indeed, if the diameter was a shade under 10 (9.55 upwards) and rounded up the circumference could be anything from exactly 30 to 31.41. Bizarrely the Greek Septuagint translation (1 Kings 7:10, LXX) written during the peak of Greek mathematics rounds or reports incorrectly the circumference as 33 cubits whilst leaving all other measurements the same.

Reading the text more closely other solutions come to mind. The vessel's diameter was measured from brim to brim (the Hebrew word means a "lip", Strong's #8193) implying a slight curvature rather than from edge to edge or side to side (verse 26 describes it as being made "like the rim of a cup" or "lily flower"). This may have exaggerated the diameter of the actual cylinder body.

Furthermore, verse 26 says that the vessel's walls were a span (Strong's #2947) or handbreadth thick (about 4-5 inches). Taking both sides into account one could measure the internal circumference for the purposes of accurate volume measurement, which would make a diameter of 10 cubits less 2 spans about right. Given the common rule of thumb (excuse the additional idiom from the human hand!) that a cubit was 18 inches and a span 4-8 inches (closed fingers or stretched thumb to little finger, well mine is) then the internal diameter would be 9 full cubits and 10/18 cubit (18 minus 2x4 = 10). In decimal terms this is 9.55 recurring which multiplied by Pi is 30.

In fact, this explanation, above, was first put forward by Rabbi Nehemiah, the writer of the earliest known Hebrew geometry textbook, Mishnat haMiddot around 150 A.D.:

"Now it is written: And he made the molten sea of ten cubits from brim to brim, round in compass, and yet its circumference is thirty cubits, for it is written: And a line of thirty cubits did compass it round about. What is the meaning of the verse, And a line of thirty cubits, and so forth? Nehemiah says: Since the people of the world say that the circumference of a circle contains three times and one seventh of the thread, take off that one seventh for the thickness of the walls of the sea on the two brims, then there remain, Thirty cubits did compass it round about."

Finally, the word "line" in a "line of 30 cubits" is the Hebrew קו qâv (Strong's #6957). However, this word is actually written as קוה qâveh (Strong's #6961), one of three places where this is the case, the others being Jeremiah 31:39 and Zechariah 1:16. The Massoretic scribes noted this and wrote a corrected קו qâv in the margin. This emendation of the text in the margin was called a qere to the actual text's given reading known as kethib. Qere meant that one could "read" the marginal reading, a scribal supposition, as opposed to the kethib or "written" word.

Curiously, the exact parallel to this passage in 2 Chronicles 4:2 has the same English text but uses the correct word קו qâv. The rabbis noticed these differences and it has been pointed out that taken numerically (Hebrew uses letters for numbers) the extra letter ה "h" adds 5 to the gematria or "number count" of the word. Gematria "number" derives from the same Greek root as geometry, our very subject under consideration. קוה qâveh comes out in numbers as 100+6+5 = 111, whilst without the last letter ה "h" it is only 106. Now our 30 cubit tank multiplied by this difference 111/106 actually turns out to be 31.415, the number we would expect using Pi to 4 decimal places - the world contemporary with this biblical account only knew Pi accurately to 2 decimal places.

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