Journal of Geomancy vol. 4 no. 2, January 1980
{21}
I see that Nigel Pennick’s latest book The Ancient Science of Geomancy perpetuates the hoary old myth that since 1 Megalithic Yard = 2.72 Feet, and since the base of natural logarithms, e, is very nearly 2.72, therefore here is evidence that the ancients knew of e.
How clever of the ancients, and how stupid of the archaeologists for dubbing (not daubing) them woad-painted savages etc.
In the present article I shall call this The Myth for short (though in fact it is but one voice of the Jubjub).
John Michell had a great deal of fun with the Myth. In one stirring paragraph of his delightful phantasmagoria The View Over Atlantis he plunged his readers into the depths of mysteriecstasy with the snippet that e is connected with logarithmic spirals and natural growth. Snail shells and sunflower heads, in other words. And what a rare old game of soldiers he had by the simple device of playing with 2.72727272…, instead of plain old 2.72. But to return to the Myth.
Here is a counter-myth. It is indeed no accident that Thom’s MY equals e feet. But it isn’t because the ancients knew of either e or feet – it is because Thom’s work is wrong, its the sense that the MY is a statistical illusion. The nature of Thom’s data does not conform to certain criteria required by the statistical theory (that of Broadbent) he is using, which nature is such that if the statistical theory is erroneously applied to the data, then it will detect a spurious unit e times the unit in which the data are measured. Further, the more observations made, the closer the approach to the ideal limit e. Now, since Thom measured his circles in feet, and measured lots of them, he detected a unit of very nearly e feet, an illusion created by a statistical misapplication.
I imagine that many a reader has gulped in megalithic panic at that last paragraph. So let me reassure him that it is a myth, and that Thom has not misapplied Broadbent. I have merely made all this up, Boojum-contrariwise, a) to give a few readers a fright, b) to show how easily and plausible a myth, without detailed proof, and without corroboration from independent sources, can be made to appear, c) to illustrate a point of comparative mythology.
My countermyth fits certain facts: measurements in feet, lots of them, and it explains the occurrence of e. Mathematicians are so used to dealing with statements like: “as the number of observations tends to infinity, the spurious quantum tends to the limit e times the basic unit of the data.”, that this part of my countermyth sounds rather plausible. Finally, my myth is not obviously false. It requires considerable technical expertise to ‘debunk’ it though in fact, there is no obligation on anyone to debunk my myth, since the onus is on me (were I serious) to prove it.
If my myth were to appear in a general publication whose readers would like to believe it were true, then I have no doubt that I would in no time be quoted as having “disproved this bally Thom Nonsense”. {22}
Now let’s go back to the Myth. As quoted by Nigel Pennick it implies two things which are likely to be overlooked by the average reader in his haste to believe: firstly, that the ancients knew of the foot, and that it was exactly the same length then as now, and secondly, that they also knew all about e.
To my knowledge there is not a shred of evidence of any ancient usage of the foot. Michell’s ‘proof’ that there was is actually no proof at all, being merely a listing of a few places where he has found it to fit, approximately, and a non listing of a lot of other places where he hasn’t. Readers of JOG may remember the fun I had with a fictitious F-unit in Vol. 1, No. 4.
Secondly, it is highly unlikely that the ancients knew of e, a number which is a product of a relatively advanced calculus – hence its appearance as the base of natural logarithms, the variant (as Michell calls it) by which the logarithmic spiral is set out, and a host of problems in mechanics and probability theory. e, like pi, crops up all over the place, but, unlike pi, which has one obvious application to circles, and hence a relatively ancient origin, e has none such and is a modern construct.
Couldn’t it be that the ancients had their equivalent of calculus which was ‘lost’ until (re)discovered by Newton and Leibnitz? I suppose so, but that, along with the assertion of an ancient knowledge of the foot, requires proof from sources independent of the Myth. The simple statement of the Myth does not prove these things, any more than a simple statement of the countermyth is a disproof of Thom.
I believe the true explanation of that factor 2.72 in association with the Megalithic Yard is one of numerical coincidence. Unfortunately, this itself is the one hypothesis incapable of proof. It becomes ‘not disproved’ every time an alternative theory is disproved, or even not proved, but it never, of itself, gets ‘proved’. Readers of JOG who get agitated by my frequent screams for proof will no doubt take great delight in this paradox of hypocrisy.
Numerical coincidences can and do occur. There were 12 apostles, 12 tribes of Israel, 12 signs of the Zodiac, 12 hours in half a day, 12 chapters in Nigel’s new book, 12 pence in my pocket and 12 steps per flight down at my local station. That’s too much 12-ness for pure chance etc. etc. And why were there exactly three little kittens who lost their mittens, or seven dwarfs involved with Snow White? And can it be by chance alone that the number of little piggies (one went to market etc.) tallies exactly with the number of toes per human foot, not to mention the number of elements postulated by the Chinese? (Could it be that the four elements known to Europeans is evidence of an antediluvian race of four-toed alchemists?). Or again, take the number of mystery 153. This was, of course, the number of fishes hauled in at one go by Simon Peter in John 21:11. Why 153?
If you want the numerology of it, see Michell and Bullinger, but don’t ask me. I have no more idea of why there were 153 fishes than I have of why my step-daughter spilled 103 dolly mixtures all over the hearth rug last Thursday. But if you want to play with a numerical Snark, well, just follow me. … {23}
“There is Thingumbob shouting;” the Bellman said.
“He is shouting like mad, only hark!
He is waving his hands, he is wagging his head,
He has certainly found a Snark!”
Now, 153 fishes in the net, plus the one that got away, makes 154.
154 = 22 × 7
which is interesting because,
pi = 22/7
And if the one that got away sounds too much of a fisherman’s yarn to you, well here is my last word on the
subject: COLEL.
“Then silence. Some fancied they heard in the air
A weary and wandering sigh,
That sounded like “-jum!” but the others declare
It was only a breeze that went by.”
Lewis Carroll