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Tue, 14 Feb 2006

More approximations to pi
In an earlier post I discussed the purported Biblical approximation to π, and the verses that supposedly equate it to 3.

Eli Bar-Yahalom wrote in to tell me of a really fascinating related matter. He says that the word for "perimeter" is normally written "QW", but in the original, canonical text of the book of Kings, it is written "QWH", which is a peculiar (mis-)spelling. (M. Bar-Yahalom sent me the Hebrew text itself, in addition to the Romanizations I have shown, but I don't have either a Hebrew terminal or web browser handy, and in any event I don't know how to type these characters. Q here is qoph, W is vav, and H is hay.) M. Bar-Yahalom says that the canonical text also contains a footnote, which explains the peculiar "QWH" by saying that it represents "QW".

The reason this is worth mentioning is that the Hebrews, like the Greeks, made their alphabet do double duty for both words and numerals. The two systems were quite similar. The Greek one went something like this:

Α1Κ10Τ100
Β2Λ20Υ200
Γ3Μ30Φ300
Δ4Ν40Χ400
Ε5Ξ50Ψ500
Ζ6Ο60Ω600
Η7Π70
Θ8Ρ80
Ι9Σ90
This isn't quite right, because the Greek alphabet had more letters then, enough to take them up to 900. I think there was a "digamma" between Ε and Ζ, for example. (This is why we have F after E. The F is a descendant of the digamma. The G was put in in place of Ζ, which was later added back at the end, and the H is a descendent of Η.) But it should give the idea. If you wanted to write the number 172, you would use ΒΠΤ. Or perhaps ΤΒΠ. It didn't matter.

Anyway, the Hebrew system was similar, only using the Hebrew alphabet. So here's the point: "QW" means "circumference", but it also represents the number 106. (Qoph is 100; vav is 6.) And the odd spelling, "QWH", also represents the number 111. (Hay is 5.) So the footnote could be interpreted as saying that the 106 is represented by 111, or something of the sort.

Now it so happens that 111/106 is a highly accurate approximation of π/3. π/3 is 1.04719755 and 111/106 is 1.04716981. And the value cited for the perimeter, 30, is in fact accurate, if you put 111 in place of 106, by multiplying it by 111/106.

It's really hard to know for sure. But if true, I wonder where the Hebrews got hold of such an accurate approximation? Archimedes pushed it as far as he could, by calculating the perimeters of 96-sided polygons that were respectively inscribed within and circumscribed around a unit circle, and so calculated that 223/71 < π < 22/7. Neither of these fractions is as good an approximation as 333/106.

Thanks very much, M. Bar-Yaholom.


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