Quick links to various main topics in my blog

http://members.tripod.com/hew_frank/index.htm

... which is equally interesting.
--
simon

Curious indeed, although I found the style a bit confusing at times. For completeness, here's a summary of a discussion we had a few years ago on Cix...

Calculating The Megalithic Yard

In The Book of Hiram, authors Christopher Knight and Robert Lomas propose a method by which ancient engineers may have calculated the Megalithic Yard. Here is a full description of the empirical method. It is based on the use of a pendulum to count 366 beats (half cycles) within a time frame defined as 1/366th of the length of a day (366 supposedly being the number of sunrises from one spring equinox to the next, as perceived by the megalithic builders). The length of pendulum cord that produces this period is then doubled to give one Megalithic Yard.

But Lomas/Knight omit a mathematical formula. It is based on the classic pendulum period (note the addition of 2 doublings - one to turn cycles into beats, and one to turn the pendulum length into a Megalithic Yard). Thus:

MY = 2(g(2T/2Pi)^2)  

where: 
   g = acceleration of gravity (approx 9.81 m/s^2)
   T = 1/366 of 1/366 of 1 day in seconds (86400/133956 = 0.644988 secs)

This produces 1 MY as 0.82699 meters, very close to the 0.82966 meters (2.72 feet) that Thom discovered. In fact it deviates by only 0.32%.

LITL.W/L
Dominic.

Hang on, I've got to do this for myself :-)

366 beats in 1/366 of a day is 366/2 = 183 periods (T) in 1/366 of a day, so the period for this pendulum is therefore:

183T = 86400/366 seconds
T = 86400/(366*183) s
T = 1.2899758 s

(I think you've used T for beats, which is why I was confused when I first read your post)

Now: T = 2 . PI . sqrt(L/g), where L is the pendulum length

So: L = g . (T / 2 . PI)^2

And: 1MY = 2L = 2 . (g . (T / 2 . PI)^2) - which is different to your formula in the coefficient of T because your T (beats) = half of my T (period)

Rearranging:
               2
          g . T       9.81 * 1.6640376
1MY =   ---------  =  ----------------  =  0.826994 m
               2         19.739209
         2 . PI

For no reason at all other than pleasing coincidence, the numerical value of 1MY in feet is only 5 parts in a thousand smaller than e :-)

86400/66978 = 4800/(61^2), which would seem to indicate that we'd be looking at a pendulum period T of 1 and 1/3 seconds if they did a Babylonia 1/360 of 1/360 division of the sky instead of a 1/366 of 1/366.

Nicely, that gives a beat of 2/3 of a second which is the only fraction the Egyptians used that didn't have a unit numerator, and had its own glyph (which evidently had associations of being sacred, for reasons now unknown).

S.

> I think you've used T for beats, which is why I was confused when I first read

Ah yes, I seemed to have confused myself in writing it up :)

Since a Megalithic circle has 366 equal parts, Lomas/Knight suggest that "when mathematics came into use in the Middle East they simply discarded 6 units to make the circle divisible by as many numbers as possible". Plausible, but I've yet to encounter a decent study on the measurement origins of arc and time.

The two earthly pillars (of upper & lower Egypt) that when linked (as an arch) with divine ma'at, ensured the physical and spiritual well being of the nation? Or perhaps the two deities (Osiris & Isis), in who's man-god offspring (Horus) the pharaoh's divine power is rooted?

LITL.W/L
Dominic.

There is a grievous error in Leroy Doggett's "Calendars" chapter from the Explanatory Supplement to the Astronomical Almanac. This same error also occurs in http://www.ast.cam.ac.uk/pubinfo/leaflets/leapyear/leapyear.html, Royal Greenwich Observatory pamphlet #48 (LEAP YEARS).

The Greenwich Royal Observatory pamphlet (henceforth G.R.O.#48) begins:

"The year is deefined as being the interval between two successive passages of the Sun through the vernal equinox. Of course, what is really occurring is that the Earth is going around the Sun but it is easier to understand what is happening by considering the apparent motion of the Sun in the sky."

"The vernal equinox is the instant when the Sun is above the Earth's equator while going from the south to the north. It is the time which astronomers take as the definition of the beginning of Spring. The year as defined above is called the tropical year and it is the year length that defines the repetition of the seasons. The length of the tropical year is 365.24219 days."

The first two paragraphs of R.G.O.#48 define the "tropical" year as the interval between vernal equinoxes and immediately thereupon give an incorrect value (365.24219 days) for the length of such a "tropical" year. The value given is not for the mean interval between vernal equinoxes but instead for a "mean tropical year" based on a "fictitious mean sun" as defined by Simon Newcomb et al. (e.g. "The elements of the four inner planets and the fundamental constants of astronomy", by Simon Newcomb, Supplement to the American ephemeris and nautical almanac for 1897. Washington, Gov't. print off., 1895.)

The length of a "real" tropical year depends upon which point of the tropical zodiac you choose to measure the year-length from (John Dee** emphasized this point as early as 1582 A.D.), and thus Newcomb's formula (or any updated version using atomic or dynamical time) cannot give you the vernal-equinox year since such formulae give an average over all points of the tropical zodiac! Values like these (365.2422 to the nearest ten-thousandth of a day) from such formulae have no more to do with the vernal equinox than they have to do with the fall equinox or for that matter the summer solstice or winter solstice or any tropical zodiacal point in-between! I have found only one modern analysis of the solar calendar which admits this fundamental fact ("Astronomical Appreciation of the Gregorian Calendar", 1949, in volume 2, #6, of Richerche Astronomiche Specola Astronomica Vaticana, by J. De Kort S.J.) but even here the value given for the length of the vernal-equinox year is incorrect (the jesuit De Kort comes up with 365.2423 days).

The result of these errors in current astronomical texts is to continue a centuries-long cover-up of the true value of the vernal-equinox year! This has importance for all Christian churches, all Persians (a.k.a. Iranians), and thus all historians of astronomy and calendars, since the major solar calendars are ostensibly deliberate attempts to keep the vernal equinox on the same date (or, in the Persian case, nearest the same midnight) of each year.

COMPLETELY ERRONEOUS ANALYSES of calendar accuracy IN GENERAL REFERENCE WORKS have resulted from this continuing scandal, e.g. SCIENTIFIC AMERICAN, May 1982, "The Gregorian Calendar"; THE DICTIONARY OF SCIENTIFIC BIOGRAPHY, p.324, in the article on "Al-Khayyami" a.k.a. Omar Khayyam; and THE ENCYCLOPAEDIA IRANICA, vol IV, pp670-672, on the Jalali and current Persian Calendar. Many works of scholarship such as these come to totally wrong conclusions about solar calendars, their accuracy, their possible reform and political history, because of pronouncements like this by the R.G.O. and other modern astronomical institutions.

Source, and continuation of the above: http://www.hermetic.ch/cal_stud/cassidy/err_trop.htm referred there by http://en.wikipedia.org/wiki/Tropical_year

Copy saved locally for archive.
--
simon

Not that that doesn't mean it may not also be extremely convenient for $TPTB, of course.

--
simon

LITL.W/L
Dominic.