The Bible and PiAuthor(s): Michael A. B. Deakin and Hans LauschSource: The Mathematical Gazette, Vol. 82, No. 494 (Jul., 1998), pp. 162-166Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/3620398 .

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THE MATHEMATICAL GAZETTE

The Bible and Pi

MICHAEL A. B. DEAKIN and HANS LAUSCH

It is widely, almost universally, believed that the Hebrew Bible gives the value of 7r as the crude approximation 3, a value much less accurate than those adopted by other ancient civilisations, such as the Babylonian, the Egyptian and the Chinese. In the English of the Authorised Version, the biblical source goes:

'And he made a molten sea, ten cubits from one brim to the other: it was round all about, and his [i.e. its] height was five cubits: and a line of thirty cubits did compass it round about.'

This is from I Kings 7:23, but it is repeated in a later verse (II Chronicles 4:2).

'Also he made a molten sea of ten cubits from brim to brim, round in compass and five cubits the height thereof; and a line of thirty cubits did compass it round about.'

A few observations are in order before we proceed.* First, there are some apparent differences of wording between the two versions. These arise because the English translation here employed was produced by a committee; they are not to be found in the Hebrew original. (We shall see later on that there are subtle differences in the Hebrew versions, but the principal one is not, nor could it be, reflected in the English.)

Second, the obvious interpretation is that the biblical author used a value of 7i. The underlined word line is quite often rendered as 'circumference', or 'cord' (more precisely it means 'tape measure' or else the measurement produced by means of a tape measure) and so the ratio of the circumference (30 cubits, about 15m) to the diameter (10 cubits 'from brim to brim', about 5m) is clearly 3.

It is thus widely held that the Hebrews of this era (around 950 BCE, the time of King Solomon) used the valuet : = 3. This belief has however been queried, and we now turn to the rationale for an alternative view. This is also presented in [1].

Belaga [1] gives some background. The molten sea was a large, bronze reservoir or tank set on the backs of twelve bronze oxen and placed in the court of the first temple. It was cast in metal which was molten during the actual casting. However, once it was cast, it held water (not molten metal). With the dimensions as given, its capacity would have been about 45000

* Quite puzzling is the fact that in the Septuagint (the earliest Greek rendering of the Hebrew Scriptures) the Kings (but not the Chronicles) passage has the circumference as 33 cubits! There are even cases of fundamentalist Christians claiming that the value 3 must be correct as it has been divinely revealed! House Bill No. 246 (1897) of the Indiana State Legislature has been so interpreted by some (although the full truth is considerably more complicated). It narrowly failed to pass.

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litres. It was 'one of the greatest engineering works ever undertaken by the Hebrew nation' [1] and its size compares with that of 'some of the largest church bells cast in modem times' (as quoted in [1]). It is certainly very dubious that an engineering work on this scale could have been carried out by a people who genuinely believed that n = 3 (although it is quite possible that no specific value was used, but rather scale drawings, mechanical instruments and the like).

However, the suggestion is that they did have a very accurate value for :n, and one that is encoded in the original Hebrew of the very passages quoted above. Posamentier and Gordan [2] state that this proposal was first put forward about 200 years ago by Rabbi Elijah of Vilna,* i.e. Elijah ben Solomon Zalman, the Gaon of Vilna (1720-1797), one of the great modem Jewish biblical authorities, whose writings also addressed geometric and trigonometric topics. However, Belaga [1] tends to attribute it to Rabbi Matityahu Hakohen Munk, either from independent research or else as an agent in the transmission of tradition. (Belaga was aware of the attribution to the Gaon of Vilna, but was unable to find any relevant passage in his writings.)

The key to the suggestion is the Hebrew word for line, occurring in the text of both the passages given above. In the original Hebrew, the two passages are almost identical, the principal difference being in this word. (There is just one other, minor, difference in the original wording. As far as we know, no significance has been attributed to this.) The first passage is traditionally attributed to the prophet Jeremiah (c 600 BCE). The second was copied from it by the scribe Ezra following the end of the Babylonian exile.

In Figure 1, the left-hand illustration gives the Hebrew word as it appears in 1 Kings 7:23; on the right is the form in II Chronicles 4:2. We see that the later version omits the final letter. The earlier version is spelt as Qof, Vav, He, that is to say QVH. (Vowels are not normally written in Hebrew. The letter Q is a rough equivalent to the English K.) In the second version, the final He (H) is omitted. This accords with a tradition under which, the earlier, written, QVH would actually be read as QV. Ezra wrote the word as it was meant to be read.

FIGURE 1: Two different renderings of the Hebrew word for line (circumference). To the left, the literary form as used in I Kings 7:23; to the right, the 'reading form' rendered explicitly in II Chronicles 4:2. (Illustration taken from [2].)

* Vilna is now known as Vilnius and is the capital of modem Lithuania.

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THE MATHEMATICAL GAZETTE

We thus have two different versions of the word for circumference,* and this leads on to the interesting part of the story. In Hebrew there is a technique for writing numbers as words; it goes by the name 'gematria'. (The ancient Greeks used a similar convention in number representation.) According to this, the letter Qof has the numerical equivalent 100, Vav the value 6 and He the value 5. Thus the word we have rendered in English transliteration as QVH has the numerical meaning 100 + 6 + 5, i.e. 111. The other version, lacking the H, reads as 106.

If we now form the ratio lI and multiply it by 3, the 'surface' or 'apparent' value of jr, as given explicitly in the text, the result is r-, a value of ;r accurate to about I of a percentage point!

Thus far we have followed essentially the account by Posamentier and Gordan [2] (but embellishing it from Belaga's fuller version [1]). What follows uses Belaga's further and more detailed exploration of the matter.

In the notation of [3] the simple continued fraction expansion for 7 is

[3, 7, 15, 1, 292, 1, ... ] and its convergents are:

22 333 355 to{ = 3, zl = --, 16 =3 = , 3t = -,

7' 106 T

113

103993 104348 74 - 33102 7 = 33215 etc.

The Rabbinical value for z is thus 72.- The surface meaning of the text gives the value Jz0, but this is deceptive; those in the know (so the story goes) see hidden in the text the much more accurate value zJr.

Now either the Rabbinical tradition is responsible for 2), and the author of 1 Kings surreptitiously coded into his text an extremely accurate value of zr, or else we have a most remarkable numerical coincidence. Which is it?

The question is not one susceptible of being decided absolutely one way or the other. We incline to the view that there is a most remarkable coincidence at work here and that it has no significance beyond this. Of course not everyone will agree with us.

Here are our reasons. First, there is the question of how we are to know that a cipher exists and

why we are to choose this particular method of decoding. There are many instances of spurious ciphers being 'decoded'; see [4, pp. 748-751]. Perhaps the most extended such is a use of numerical equivalents of alphabetical characters and other similar techniques to 'prove' that Queen Victoria was the true author of Tennyson's In Memoriam [5]. (This latter was produced as a satirical attack on the Baconian theory and played a considerable part in discrediting it.)

Next up, the relative error in Jr2 is, as we have seen, about ? of a percentage point. That of mr3 is much, much smaller, being less than one * Note that the Septuagint however uses two distinct words in the two versions.

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hundred-thousandth of one percentage point! However, the denominators are very close to one another. So if we want a good approximation to nT, we might as well not bother with z2, but instead go immediately to r3.

Now there is a most remarkable numerical coincidence in the value of 7r3, i.e. 355. If we start at the bottom left of this fraction and follow the digits round in the pattern of the letter S, we find the simple and memorable pattern (pointed out to us by Dr Russell Smith of Australia's CSIRO): 113355. This pattern clearly holds no great significance, although it makes for a highly accurate and easily memorable approximation to st, one that deserves to be more widely known. We incline to the view that we should see the appearance of 7r2 in a similar light.

If more need be said, then it could perhaps be found in other places in the Hebrew Bible. The word here represented as 'line' is to be found in its QVH form at Jeremiah 31:39 and at Zechariah 1:16, and in its QV form in many other places. In all these cases, however, there is no ready reference to circular measure, the 'line' or 'tape measure' is stretched out straight or else is more complicated than simply circular (probably two conjoined straight line segments). The type of analysis Belaga applies to I Kings 7:23 could also be applied to Jeremiah 31:39 or to Zechariah 1:16, but it is a little hard to see how the ratio I1I could be given any significance in these different contexts.*

However, there will be those who see the biblical pattern as significant, and certainly they have a case, even a strong case. Nonetheless, we should remember the words of Aharon ben Zalman Emmerich Gumpertz (1723- 1769) who, having been secretary to Pierre Moreau de Maupertuis, the President of the Berlin Academy, observed: 'Mathematics and the sciences have no business with the divine religion'.t

Acknowledgements For making Reference [1] available to us, we thank M. Closs (for the

published version) and M. McKinzie (for an electronic update). B. Rechter provided a copy of the relevant page of a concordance to the Hebrew Bible.

References 1. S. E. G. Belaga, On the rabbinical exegesis of an enhanced biblical

value of zr, Proc. XVIIth Can. Cong. Hist. Phil. Math., (1991) pp. 93- 101.

* The Authorised Version translates these texts: 'And the measuring line shall yet go forth over against it upon the hill Gareb, and shall compass about to Goath.' and 'Therefore thus saith the LORD; I am returned to Jerusalem with mercies: my house shall be built in it saith the LORD of hosts, and a line shall be stretched forth upon Jerusalem.' In the case of the former, the Revised Version is more explicit: 'And the measuring line shall yet go out straight onward unto the hill Gareb, and shall turn about unto Goah.'

t From "Ma'amar hamada' " ["The science treatise"] in Megale sod (Hamburg, 1765).

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THE MATHEMATICAL GAZElTT'E

2. A. S. Posamentier and N. Gordan, An astounding revelation on the

history of r, Math. Teacher 77(1) (1984) pp. 52, 47.

3. C. D. Olds, Continued fractions Yale University Press, New Haven (1963).

4. D. Kahn, The codebreakers Weidenfeld and Nicolson (1967).

5. R. A. Knox, The authorship of 'In Memoriam'. In Essays in Satire (2nd edition), Kennikat, Port Washington, NY (1968).

MICHAEL A. B. DEAKIN

HANS LAUSCH

Department of Mathematics, Monash University, Clayton, Vic. 3168, Australia

e-mail: Michael. Deakin @ sci. monash. edu. au e-mail: Hans. Lausch @ sci. monash. edu. au

Um, er, pass the dictionary Can you remember the sine and cosine rules, or what Pythagoras theorum is?

Written for the general reader, this dictionary contains the essential facts about mathematics, alphabetically arranged to put the facts at your fingertips.

Seen by Frank Tapson on the dust-jacket of the Hutchinson Dictionary of Mathematics. The dictionary entry does not repeat the spelling error.

Keep death off the roads, drive on the pavement The facts about speeding are clear. At 20 mph, one in 20 pedestrians are killed,

at 30 mph half are killed and at 40 mph only one in ten will survive. Driving more slowly can reduce the severity of a casualty in that awful situation.

This report in the Mid Devon Advertiser, 19 September 1997, carried a serious message, but failed to make it clear to Frank Tapson that the figures refer to pedestrians invoved in accidents, rather than all pedestrians.

Frustrating for mathematics teachers too The RAF fighter pilot Andy Green drove his Thrust SSC car on the fastest

officially timed run in land speed racing history, but missed breaking the sound barrier by a frustrating 0.003 per cent of the speed of sound. ... Timing officials gave the run a provisional Mach 0.997.

Yet more confusion over percentages, this time in The Times, 14 October 1997, spotted by Frank Tapson.

Law of diminishing returns? Beware of special offers. Not all two for the price of three deals are as good

value as they seem. Prima magazine's advice on how to cut down on impulse buys (November 1997)

did not impress Christine Brooks.

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