I’ve been working for some time now on the problem of ancient geodesy – essentially, if ancient people found how large the world was earlier than they are usually given credit for, what did they think was the right way to reckon this and express it?
It came to light that there are several ways this can be done – geodetic modelling is one of them – that is modelling the circumference, diameter, or radius of the earth at ratios like 1 foot = 10^n miles, so for example to express a 24901.19742 mile equatorial circumference, someone might design a pyramid 249.0119742 ft high, or 249.0119742 ft long on each side.
24 of the standard Remen can be used for geodetic modelling as the cube root of 24901.19742. (24 x 1.216733603 ft)^3 = 24901.19742.
Another way is geodetic measurement, units of measurement that can be used effectively to measure the earth and generate what are ideally memorable numbers of units.
108 x (10^n) Remens of 1.216733603 feet = 131407229.1 feet = 24887.73279 x 5280.
24887.73279 is near to 24883.2, a convenient consensus figure for the mean circumference of the Earth (used by both myself and authors like John Michell), so the familiar standard Remen value can be used for actual measurement of the earth on this basis, given slight corrections to the mean circumference and mile values.
Things of course become more complex when we go to ponder whether the ancients were able to distinguish the polar circumference from the equatorial circumference. I’ve been working with such figures for years but they didn’t come with instructions. If we want to keep a nice round number of units like 108 x (10^n) but the circumference value changes like that, do we change the circumference value, the number of units in it, or change the unit to keep root values like 108 x (10^n) constant even while the circumference is changing?
Recently I might have had a breakthrough with this question, going over some data on Stonehenge’s sarcen circle. The mean values for the sarcen circle require a very slight adjustment to become a valid figure, which causes the ratios between maximum and mean, and between mean and minimum, to become slightly different from each other.
This is advantageous as it provides us with an impressive pair of figures for these ratios, but there may have been even more to the grouping of these figures together that way than I had realized.
I’ll skip a review of the sarcen circle basics as I’ve written lots on Stonehenge here not that long ago, but basically for the sarcen circle,
Max / mean = 1.032795559
Mean / Min = 1.033542556
To be specific about the math in question then, it looks like this, as described in these notes from several days ago
(1 / 1.033542556) x 1.067438159 = 1.032795556
1.032795556 x 1.067438159 = 1.102445487
That is a different thing than 1.067438159 / 1.177234772 = .9067249892 = 1 / 1.102870233 = 2.720174976 / 3
1.102445487^2 = 1.215385832
1.215385832 x (108 x (10^n)) = 131261693.6 = 5280 x 24860.16924
Thus 1.215385832 is remarkably close to lifelike figure for a “polar geodetic measurement Remen”.
These even more recent notes hint at the possible identification of an “equatorial geodetic measurement Remen”
Geometry may mandate the existence of a Remen of 12 / (Pi^2), as may the astronomical associations between the Remen and Solar Year.
Solar Year 3600 / (Pi^2) = 364.7562611; 364.7562611 / 300 = 1.215854204 = (12 / (Pi^2)), even if this is NOT the Remen of choice for most purposes, and may only be good for diagonal measurements relative to other better established unit values.
1.215854204 x (Pi^4) = (12 x (Pi^2)) / 10 = 11.84352528, which seems to be able to serve as the link between Half Venus Cycle and Venus Orbital Period.
1.215854204 x (Pi^5) = 3720.753202, half of the inner sarcen circle area of Stonehenge
1.215854204 x (Pi^6) = 1168.909092 = 292.2272731 x 4
Normally this is an affront to the 24 Remen value of 24 x 1.216733603 = 292.01760646 because cubing 292.01760646 gives an equatorial circumference value as accurate as 24901.19742 vs textbook value ~24901.5, whereas cubing 292.2272731 gives 24955.26790, adding almost 55 unwanted and unnecessary miles onto the equatorial circumference, which probably isn’t acceptable even by ancient standards.
However, 292.2272731 / 24 = 1.217613638
This may be the missing “equatorial measurement Remen” – a unit that can be used for actually measuring the world rather than simply modelling it across ratios like 1 foot = 1 mile x 10^n.
1.217613638 x 108 x 10^n = 131502272.9 ft = 5280 x 24905.73350
And that may be about as close as its going to get using units like these?
There is an alternate equatorial circumference figure as well, of 24903.44232 miles that I’ve seemingly found at both Giza and Tikal, with special focus on correct context at Tikal. Depending on how strong they really are as numbers, 1.217589025 or 1.217592278 might have been used with the ~24903 mile figure.
Since 1.217589025 is the value that formulaically gives the familiar standard Royal Cubit as the requisite modified mile value / (2^n), one might have to guess that this is the correct one.
Thus we would have in total as our possible geodetic Remens for actual measurement
1.215854204 x 108 x 10^n = 131312254.0 ft = 5280 x 24869.74508 = ~24859.73400 miles
1.216733603 x 108 x 10^n = 131407229.1 ft = 5280 x 24887.73279 = ~24883.20000 miles
1.217589025 x 108 x 10^n = 131499614.7 ft = 5280 x 24905.23006 = ~24901.19742 miles
1.217613638 x 108 x 10^n = 131502272.9 ft = 5280 x 24905.73350 = ~24903.44232 miles
It may be possible to do better with some of these figures, but for now this represents the first complete set of candidate figures for Remen values suitable for geodetic measurement.
Work is still in progress on a set of candidates for Royal Cubit values with similar functions in geodetic measurement and the appropriate root values.
Note that herein we not only see that the standard Remen value of 1.216733603 has the distinction of being useful for both geodetic modelling and geodetic measurement. One candidate scheme for geodetic Royal Cubit values affords similar distinction to the standard Royal Cubit value of 1.718873385 (the Morton Cubit), which seems very promising.
Meanwhile, here again is Jim Alison’s paper on geodetic functions of the Remen:
The Measure of the Remen and the Royal Cubit;and the Meridian of Egypt and the Earth
By Jim Alison – November, 2019
http://home.hiwaay.net/~jalison/blu5.PDF
–Luke Piwalker
Luke Piwalker 