To the memory of John Michell (1933 – 2009)

*Nick Kollerstrom is historian of science, a former honorary research fellow in Science and Technology Studies at University College, London (UCL), and a former lunar gardening correspondent for the BBC. He is the author or co-author of a number of books, including Gardening and Planting by the Moon (an annual series beginning 1980), Newton’s Forgotten Lunar Theory (2000), ‘Crop Circles – The Hidden Form’ (2002), and Terror on the Tube (2009)*

### The *Ur* – Number

On the banks of the river Tigris, around 1900, a US archaeological team dug up tablets of ancient Sumerian arithmetic, from 2,300 BC. What they contained was astonishing – though you are unlikely to find an account of them in histories of mathematics or early science. These tablets are certainly amongst the earliest written maths on record.

On such ancient tablets, would one expect concern with such matters as, how to count sheep, or whether two plus two equals four, or how many days were there, in a lunar month? Instead, we are startled to find that they described mathematical operations centred around the number 12,960,000, the fourth power of sixty. Tables of multiplication and division were set out:

‘What in particular is the meaning of the number 12,960,000 (=60^{4} or 3600^{2}) which underlies all the main texts here treated…? The division tables from the main temple library of Nippur, which are all based upon 12,960,000 … All the multiplication and division tables from the temple libraries of Nippur and Sippar, and from the library of Ashurbanipal, are based upon 12,960,000.’ [1]

Scribes had grappled with this huge number, millennia ago. (For background on this remarkable archaeological expedition, led by the German Assyriologist Hermann Hilprecht, see here) Why does no history of science or mathematics allude to this – I tried checking out the classic books by Otto Neugebauer and Van der Waerden, the great historians of ancient mathematics [2],[3] – does it just not fit in with their ‘primitive man’ paradigm?

Hilprecht endeavoured to ‘explain’ the interest in this huge number by quoting Plato. But, that was a millennium and a half later … Plato did indeed get to hear of this huge number – from Babylon, experts believe [4], plus maybe some stories of its importance. What did he make of it? He interpreted it as denoting a cycle of time, the so-called ‘Platonic Year’ of 36,000 years.[5] If, he explained, the year be taken as 360 days, then that Number would equal the days in this huge period.

No such astronomical period exists, but it gets worse. Plato *philosophises *about good versus bad births –and some people reckon he is here being a bit astrological – depending upon the number of days of the gestation period: whether that number of gestation divides into the primal, *ur*-number, or not. He called the big number, ‘Lord of Better or Worse Births,’ also it gets alluded to as Plato’s ‘nuptial number.’

Hipparchus, credited with being the first astronomer to discover the precession of the equinoxes, estimated it as one degree per hundred years. Clearly, that would give Plato’s Period for the overall precessional cycle. Later on, Claudius Ptolemy in Alexandria *designated* the precessional period – using the same value as Hipparchus – as ‘the Platonic year.’ We may doubt whether Plato knew anything about precession, and certainly his treatment of the big number, *the fourth power of 60*, had nothing to do with it.

### Earth-Measure

A couple of millennia roll by, and gradually the size of the Earth starts to be known; whereby the subject of geodetic metrology can arise: this examined indications that the ancient units of measure had originally come from the size of the Earth. (The endeavour to define the metre geodetically, around 1800, in terms of a polar meridian passing through Paris, such that a Great Circle around the Earth would be 40,000 km, was nearly right, it turned out to be 40,008.6 km.) The British Platonist philosopher John Michell gained especial insight into this endeavour, and he wrote:

The Greek foot is the most obviously geodetic of all units. In the earth’s meridian circumference … there are 129,600,000 Greek feet. 1,296,000 is the number of seconds in the 360 degrees of a circle… [6]

The *Ur*-number has here reappeared – but neither John Michell nor any of the other Earth-metrologists (if we want to call them that) twigged that early mathematics had in fact come across this big number, or that it once had a central meaning.

What we might call the *main-sequence* of British Earth-geodesy books – Nicholson, *Men and Measures* (1912), Berriman *Historical Metrology* (1956), Ivimy *The Sphinx and the Megaliths* (1976), and Michell *Ancient Metrology* (1981) – all made a similar comment upon the Greek foot as being the primary, geodetic unit: and found themselves having to burden their readers with the hard-to-believe concept that some culture way back in prehistory had divided up the Earth’s circumference into degrees, minutes and seconds: ‘possibly the circumference was rated 60^{4} units in remote antiquity,’ opined Berriman (p.19). Both Nicholson and Ivimy discussed how the Greeks had inherited a base-60 system of length measure even though this was foreign to their maths: ‘At least sixty centuries ago the Chaldean astronomers had divided the circumference of the Earth, and of circles generally, into 360 degrees (that is 60 x 60) each of 60 parts.’ (Nicholson p.15, see also Ivimy p.55)

All of these authors postulated a definition of the ancient Greek foot comparable to that of the nautical mile in the 19^{th} century, such that there would be 6000 Greek feet in a nautical mile. To quote Wiki (N.Mile),

‘Both the Imperial and US definitions of the nautical mile were based on the Clarke (1866) spheroid: specifically, they were different approximations to the length of one minute of arc along a great circle of a hypothetical sphere having the same surface area as the Clarke spheroid.’

It was agreed to define the Nautical Mile as 6080 feet.

Francis Bacon affirmed in his *New Atlantis,* ‘You shall understand (that which you will scarcely think credible) that about three thousand years ago, or somewhat more, the navigation of the world (especially for remote voyages) was greater than at this day.’ [7] If, as Graham Hancock has argued, we are ‘a species with amnesia,’ [8] then we may in some degree hope to overcome that amnesia by grappling with the Earth-measure question. Did nations long ago use such a measuring system, one assuming a *mean *circumference? Only a maritime civilisation would have needed that.

Primary evidence here comes from the Acropolis length & breadth as measured very carefully by Francis Penrose in 1882 (100 x 225 Greek feet): his measurements gave a mean value of 100 Greek feet as 101.368 feet; while taking a modern mean Earth radius, and dividing its circumference into arcseconds, gave 101.365 feet. [9]

Visiting Athene’s temple at the Acropolis (constructed about ten years before he was born), Plato never apprehended that its width, of 100 ‘Olympic’ Greek feet, derived from this very number, – that indeed it provided the most precise measure in existence of one 60^{4} fraction of Earth’s *mean circumference*. Only in our time, as we come into high-precision knowledge of Earth’s dimensions, can this be ascertained. Whoever built the Acropolis had to have known this – or come from a tradition which did. An amnesia prevailed amongst Greek philosophers over this matter, of how ‘geo-metry’ had once signified earth-measure. Visiting the stadium racetrack in Athens, Plato would indeed have mulled over its six hundred feet length, and where, he would have wondered, did that unit, the number of feet in a stade, come from? The answer, that this Greek stade measured out one tenth of an arcminute of a Great Circle around the Earth, could alas never dawn upon him. ‘600 Olympic Stades were thus equal to one degree of latitude, and 100 Greek feet equalled 1 second of latitude – facts of which the Greeks themselves were quite unaware’ wrote John Ivimy. [10]

It never dawned upon Plato, leaning against the newly-built Acropolis, that it measured the Earth. No Greek philosopher twigged that their system of units was Earth-based. He took the* Ur*-number 60^{4} to be a measure of Time, not Space, and garbled the ancient wisdom with a nonexistent period; only sorted out in our time by the Platonist philosopher John Michell.

### Postscript

The book *The Measure of Albion, the Lost Science of Prehistoric* *Britain* by Robin Heath & John Michell (2004) has the basic length-measures of different cultures in antiquity all derive from one value of the Earth’s circumference, which they give as 24,883.2 miles (pp. 6, 12, 20 etc) and they called that the ‘Meridian circumference’. (By that term they meant a Great circle passing through the poles, which I don’t believe it ever was). They obtained the number from Michell’s earlier work *Ancient Metrology,* where it’s given as the fifth power of 12. Wiki gave 24,880 miles as a modern, *mean* Earth-circumference, so it’s as close as makes no difference.

Let’s write the (rather mystic-looking) equations, as John Michell saw them:

Mean Earth-circumference = 12^{5}/10 miles = 60^{4} x10 Greek feet.

That works as exactly as can be ascertained by the figures. From that ratio there follows John Michell’s insight [11] that

English mile / Greek mile = Greek mile / Roman mile = 25 / 24

It just tumbles out of the maths so to speak; bearing in mind that the Greek & Roman miles had 5000 feet whereas the English mile had 5280 feet. The ratio linking Greek and Roman feet or miles by 25:24 is well-established and was accepted by ancient sources; whereas that linking the English and Greek miles is not, and seemingly has no business to exist. After all, it was the Roman and not the Greek mile that was introduced into Britain and there was ‘no trace of the Olympic foot in Northern Europe’ [12].

By Nick Kollerstrom, MA Cantab., PhD, FRAS (for other articles by NK, see here)

### Endnotes

- Herman Hilprecht,
*The Babylonian Expedition of the University of Pennsylvania*, Series A, Vol. 20,*Mathematical, Metrological and Chronological tablets from the Temple Library at Nippur*, Philadephia 1906, pp.31-2. [back to text] - See, eg, ‘The Sixty System of Sumer’ by A. Seidenberg,
*Archive for History of the Exact Sciences*, 1962, 2,5, pp.436-440, communicated by Van der Waerden. [back to text] - George C. Joseph’s highly-praised
*The Crest of the Peacock, non-European roots of Mathematics,*2011 also omits their mention – even while alluding to early Sumerian math tablets from Nippur (pp.131-134), also a 4^{th}millennium BC tablet from Uruk that found areas of fields. Field-area computation is the sort of thing a historian*might expect to find*on such tablets; which cannot be said of 60^{4}– based computations. [back to text] - J.Adam,
*Republic of Plato,*1902, Vol. II (Book VIII), p.201-9 and 264-306. [back to text] - Richard Dumbrill, ‘Four Mathematical texts from the Temple Library of Nippur’,
*Journal of the American Oriental Soc*. Vol 29, 1908, pp.210-219. [back to text]

- Robin Heath & John Michell,
*The Measure of Albion**The lost science of prehistoric Britain,*2004, p.80 (Chapter written by John Michell). Strictly the 60^{4}division of Earth’s circumference gives 10 Greek feet, see equation below. [back to text]

- J. Spedding,
*The works … of Francis Bacon*, 1858-74, Vol. 3, p.140-1. [back to text] - Graham Hancock,
*Fingerprints of the gods,*1995, p.199. [back to text] - N.K., ‘The Acropolis Width and Ancient Geodesy’
*,**Cal Lab the International Journal of Metrology*, 2005 Vol 12, pp.38-41. [back to text] - John Ivimy,
*The Sphinx and the Megaliths*, 1976, p.55 [back to text]

*The Measure of Albion,*p.79. [back to text]- Edward Nicholson,
*Men and Measures,*1912, p. 49. [back to text]