Parthenon as a New Model of the Meridian

This was published as The Geodetic And Musicological Significance Of The Shorter Side Length Of The Parthenon As Hekatompedon Or ‘Hundred-Footer’ in Music and Deep Memory: Speculations in ancient mathematics, tuning, and tradition, in memoriam Ernest G. McClain. Edited by Bryan Carr and Richard Dumbrill. pub: Lulu. photo: Steve Swayne  for Wikipedia on Parthenon.

This note responds to Kapraff and McClain’s preceding paper, in which they discover a many-faceted musical symbolism in the Parthenon. Specifically,  Ernst  Berger’s  new measurements include the shorter side of the triple pedestal of the monument as an accurate length to represent one second of the double meridian of the earth. By applying a knowledge of ancient metrology, Anne Bulckens’ doctoral derivations of a root foot can resolve to a pygme of 9/8 feet, of which one second of latitude would contain 90 such feet. However, as a ‘hundred footer’, the foot  length  should  then be 81/80 (1.0125) feet, the ratio  of  the syntonic comma. This would indicate a replacement, by Classical times, of the geographical constant of 1.01376 feet  within the model of the earth since the original model, by the late megalithic, assumed that the meridian was exactly half of the mean circumference of the earth. These alternative geographical constants co-incidentally represent the ubiquitous theme in ancient musicology of the transition between Pythagorean and  Just tunings and their respective commas of Pythagorean 1.01364 … (in metrology 1.01376) and syntonic 81/80 (1.0125).

By Classical times the term hekatompedon or ‘hundred-footer’ had evolved, to describe the ideal dimensionality of Greek peristyle temples. One of the earliest, the Heraion of Samos, came to be 100 feet long by the end of the 8th century[1], in contrast to the surface width of the Parthenon’s stylobyte which had been established as in the range 101.141 (Stuart, c.1750) to 101.341 (Penrose in 1888) feet[2].

Recent measurements in 1982 by Ernst Berger[3] found that the top surface of the stylobyte was just over 101.25 feet wide4 and that the most frequently occurring length was 857.6 mm. Anne Bulckens’[5] corresponding foot measure for this would be a step of 2.5 feet, each of 9/8 (1.125) feet, to within
one part in 2500; a foot length called a pygme within historical metrology, after the size of small men first mentioned when Herakles was travelling back from India6. The shorter ends of the Parthenon’s stylobyte would then be 90 such feet across.

However, should the two ends be divided by 100, the required foot length of 101.25 feet becomes a microvariation of the English foot, namely 81/80 (1.0125) feet, a ratio identical with the syntonic comma. This is another ratio crucial to the history of ancient tuning theory; being found between pure Pythagorean tones (9/8) and their counterparts within just tuning (10/9); when string lengths are given specific whole number lengths to specify their pitches intellectually.

1. Hurwit, Jeffrey M., (1987), The Art and Culture of Early Greece, 1100-480 B.C., Cornell: Ithaca, 74-77
2. Berriman, A.E., (1953) Historical Metrology, London:
Dent. IX, 116-120.
3. Berger, E., ed. (1986) Parthenon-Kongress Basel, 2 Vols, Mainz: Philipp von Zabern.
4. an average noted by Berriman, 119.
5. Bulckens, A.M. (1999) The Parthenon’s Main Design Proportion and its Meaning, [Ph.D. Dissertation], Geelong: Deakin University, 269 pp. ; (2001) The Parthenon’s Symmetry in Symmetry: Art and Science (Fifth Interdisciplinary Symmetry Congress and Exhibition of the ISIS-Symmetry), (Sydney, 2001), no. 1-2, pp. 38-41.
6. Philostrates of Lemnos (c. 190 – c. 230 AD) Imagines Heracles among the Pygmies, see Loeb Classical Library

A recent article by Jay Kapraff and Ernest McClain[7] observes that the width of the Parthenon symbolically defined one second of latitude (taking surface lengths as linear fractions of latitude). This implies the double meridian length was known within 0.003% of its modern estimation.

A geodetic symbolism was apparently given to shorter side length of the Parthenon, making it smaller than it would have been if modelled on the circumference of the earth as one 3,600th of one 360th part of the mean earth. If so, this geodetic meaning of the Parthenon can be compared with monuments built two thousand years earlier, such as Stonehenge and the Great Pyramid of Giza, within which the relationship of the mean earth was specified, relative to the polar radius, using the same metrological system.

The ancient model of the earth, recovered[8] by John Neal[9] and John Michell[10], used three different approximations of π to model the distortion of
the rotating planet relative to its mean, or perfectly spherical, size. In that model, the Meridian was assumed to be half the circumference of the mean earth of 44 times 126 (131,383.296) feet or 24,883.2 miles. Had the Parthenon’s builders used this model then its ends would be 101.376 feet in width and one hundredth of this would be a foot of 1.01376 feet, the foot known as the ‘Standard Geographical’ Greek foot[11].

The mean circumference of the earth (24,883.2 miles) and the actual double meridian length (24,859.868 miles) are in the same ratio as the geographical foot of 1.01376 (3168/3125) and 1.0125 feet: the 81/80 foot measure that makes the Parthenon’s 101.25 feet a ‘hundred footer’. It is therefore reasonable to assume that, between the building of Stonehenge and Great Pyramid (by 2,500 B.C.) and the building of the Parthenon (designed by 447 B.C.), a more accurate
measurement of the Meridian had superseded the previous assumption, within the old model, that the Meridian was half the length of the mean earth circumference.

7. The Proportional System of the Parthenon, in preparation for the In Memoriam volume for Ernest McClain (1918-2014)
8. Michell by 1980 and Neal, fully formed, by 2000.
9. Neal, John (2000) All Done With Mirrors, Secret Academy, London.
10. Michell, John (1982) Ancient Metrology, Pentacle Books, Bristol, 1982; (2008 new ed.) Dimensions of Paradise, Inner Traditions: Rochester.

Further to this, one can see how the transition from Pythagorean to just tuning systems[12] is strangely present in the relationship between the mean earth circumference and the actual meridian length, since the geographical constant of 1.01376 is near identical to the Pythagorean comma of 1.0136433 while the (chosen) ratio of 1.0125 is the syntonic comma and this, times 100, is near identical to the actual length of one second of latitude which would be 100 times 1.0128 feet[13], just one third of an inch different from a more
modern result.

The Parthenon ‘Hundred footer’ was able to dimensionally reference one second of the Meridian by having its shorter sides one hundred feet of 1.0125 feet long. Aligned to north, this presented accurate Classical knowledge of the
Meridian’s length. The monument expresses other musicological features via its metrology: the 81/80 foot unit is 125/128 of the Athenian foot of 1.0368 feet, a musical interval called the minor diesis, also found within just intonation and equaling the deficiency of three major thirds to the octave

12 The latter prevalent in other aspects of the monument, see Kappraff, J. and McClain, E.G. (2005: Spring–Fall) The Proportions of the Parthenon: A work of musically inspired architecture, Music in Art: International Journal for Music Iconography, Vol. 30/1–2.
13 A non-harmonic 79/78 feet.

π and the Megalithic Yard

The surveyor of megalithic monuments in Britain, Alexander Thom (1894 – 1985), thought the builders had a single measure called the Megalithic Yard which he found in the geometry of the monuments when these were based upon whole number geometries such as Pythagorean triangles. His first estimate was around 2.72 feet and his second and final was around 2.722 feet. I have found the two megalithic yards were sometimes 2.72 feet because the formula for 272/100 = 2.72 involved the prime number 17 as 8 x 17/ 100, and this enabled the lunar nodal period of 6800 days to be modelled and and the 33 year “solar hero” periods to be modelled, since these periods both involve the prime number 17 as a factor. In contrast, Thom’s final megalithic yard almost certainly conformed to ancient metrology within the Drusian module in which 2.7 feet times 126/125 equals 2.7216 feet, this within Thom’s error bars for his 2.722 feet as larger than 2.72 feet.

This suggests Thom was sampling more than one megalithic yard in different regions or employed for different uses. Neal [2000] found for Tom’s statistical data set having peaks corresponding to the steps of different modules and variations in ancient metrology, such as the Iberian with root 32/35 feet and the Sumerian with root 12/11 feet. It is only when you countenance the presence of prime numbers within metrological units that one breaks free of the inevitably weak state of proof as to what ancient units of measure actually were and, more importantly, why they were the exact values they were and further, how they came to be varied within their modules. However, the megalithic yard of 2.72 appears to outside the system in embodying the prime number 17 for the specific purpose of counting longer term periods which themselves embody that prime number.

The discipline of using only the first five primes {2 3 5 7 11} must have been accompanied by the perception that only if primes were dealt with could certain ends be served. This is crystal clear when we come to musical ratios in which the harmonic primes alone are used of {2 3 5} with an occasional “passenger” of the prime {7} as in 5040 which is 7 x 720, the harmonic constant.

Using 2.72 feet to count the Nodal Period

The first remarkable characteristic of 2.72 feet is that 8 x 17 in the numerator means that the approximation to π of 25/8 = 3.125 can, in (conceptually) multiplying a diameter, provide an image of 25 units on the circumference of a stone circle. For example a diameter of 2 MY would suggest 17 MY on the circumference, which is quite remarkable. Further to this, we know that the 6800 days of nodal cycle is factored as 17 x 400 and that the MY was shown (acceptably) to have been made up of 40 digits (in conformance to the general tradition within metrology that there are 16 digits per foot and 40 for a step of 2.5 feet, which a MY traditionally is). The circumference of 17 MY is then 17 x 40 digits which means that a diameter of 20 MY would give a circumference of 17 x 400 digits equalling 6800 digits as a countable circumference in digits per day.

This highlights how prime number factors played a role, in the absence of arithmetical methods, in solving the astronomical problems faced by the late stone age when counting time periods in days.

Developmental Roots below 6

Square roots turn out to have a strange relationship to the fundaments of the world. The square root of 2, found as the diagonal of a unit square, and the square root of 3 of the diametric across a cube; these are the simplest expressions of two and three dimensions, in area and volume. This can be shown graphically as:

The first two roots “open up” the possibilities of
three-dimensional space.
Continue reading “Developmental Roots below 6”

The Megalithic Pythagoras

Pythagoras of Samos (c.600BC) very likely gleaned megalithic number science on his travels around the “Mysteries” of the ancient world. His father, operating from the island of Samos, became a rich merchant, trading by sea and naming his child Pythagoras; after the god of Delphi who had “killed” the Python snake beneath Delphi’s oracular chasm, now a place of Apollo. The eventual disciples of Pythagoras were reclusive and secretive, threatening death on anybody who would openly speak of mysteries, such as the square root of two, to the uninitiated. It can be seen from the previous post that many such “mysteries” were natural discoveries made by the megalithic astronomers, when learning how to manipulate number without arithmetic, through a metrological geometry unfamiliar to the romantic sacred geometry of “straight edge and compass”.

As previously stated, the vertex angles of right triangles whose longer sides are integer in length, are angular invariants belonging to the invariant ratio of their sides. To create a {11 14} angle one can use any multiple of 11 and the same multiple of 14 to obtain the invariant angle whereupon, the hypotenuse and base will shrink or grow together in that ratio: any length on the “14” line is 14/11 of any length below it on the “11” base line and visa versa.

If one enlarges the base line to being 99 then the diagonal of the square side length 99 will be 140, which is 99 times the square root of two. In choosing, as I did, to enlarge 91 (the quarter year) to 9 x 11 = 99, I encountered the cubit of the Samian (“of Samos”) foot of 33/35 feet, as follows. When Heraclitus, also of Samos, visited the Great Pyramid he gave its southerly side length as 800 “of our feet” and 756 English feet (the measured length) needs to be divided by 189 and multiplied by 200 to obtain such a measurement, giving a Samian foot of 189/200 (=0.945 feet) which is 441/440 of the Samian root foot of 33/35 feet. 33/35 x 3/2 = 99/70 (1.4143) feet but its inverse of 35/33 x 4/3 = 140/99 feet.

There is then no doubt about Samos as being a center in the Greek Mysteries since, the form of the Greek temple seems first to evolve there. For example, 10,000 feet of 0.945 feet equal 945 feet, the number of days in 32 lunar months. The Heraion of Samos (pictured above) has been shown to have had pillars around a platform (a peristyle), and an elongated rectangular room (a cella), involving megalithic yards and a 4-square geometry cunningly linking lunar and solar years, to alignments to the Moon’s minimum using the {5 12 13} second Pythagorean Triangle. (diagram at top is from figure 5.9 of Sacred Geometry: Language of the Angels).

The reason for the Samian (lit. “of Samos”) foot being 33/35 feet appears to be that as a cubit of 99/70 feet, or √2 =1.4142, it is the twin of 140/99 as 1.41. In the geometrical world such foot ratios were exact, relative to the English foot; which is the root of the Greek module and of all other rational modules, such as the Royal of 8/7 feet. Such cubits could measure across the diagonal the same number as the side length in English feet. Such measures became essential for building of rectangular temple structures in Greece and further east, but when the metrological geometry, of square and circle in equal perimeter, was the focus, 140 in the diagonal can use 99 in the base (or side-length of the square).

If we remember that the 99 length must be rooted from the shared center of the square and equal circle then, the side length of the square must be twice that, or 198. This means that the perimeter of the square must be 4 times that, equal to 792, at which point readers of John Michell’s books on models of the world will recall that the diameter of the mean earth can be presented, within an equal perimeter design, if each unit is multiplied by 720 units of 10 miles, my own summary being in my recent Sacred Geometry book , chapter 3 on measuring the Earth. This model Michell called The Cosmological Prototype, where the mean earth diameter is (quite accurately) 7920 miles.

If the square of 198 feet is rolled out into a single line, it “becomes” the mean diameter of the Earth in units of 10 miles. For this sort of reason, my 2020 book was called Language of the Angels, since this model looks like a first approximation of the mean earth size which a later Ancient Metrology would improve upon as to accuracy, by a couple of miles! That is, that the earth’s dimensions follow a design based upon metrological geometry and the properties of numbers.

John Michell finalized his Cosmological Model in an Appendix to The Sacred Center, and in his text on “sacred Geometry, Ancient Science, and the Heavenly Order on Earth” called The Dimensions of Paradise, both published by Inner Traditions.

Seven, Eleven and Equal Perimeters

above: image of applications involving sacred geometry based upon pi as 22/7 and a circle of equal perimeter to a square, from a previous post.

The geometrical and other relationships between different numbers are easily found to be useful through simple experiments. The earliest approximations to pi (22/7) was key in the megalithic and later ancient cultures, for making circles of a known diameter and circumference, the foremost using the numbers 7 and 11 doubled twice. A staked rope of length seven will create a circumference of 44, to a high degree of accuracy.

But what is pi? it actually connects two different worlds, of extensive linear measure and of intensive rotational measure. As the radius rope is made larger the circle expands from its center but it remains a whole circle, except that its circumference is made up of more “units” all according to the ratio pi = 22/7, in a good approximation.

But measuring a circumference is fiddly, it is circular! In contrast, it is very much easier to work with squares since their perimeter is four times their side length. And in many cases, one does not really need to measure the perimeter. Because of this, the megalithic looked for and discovered an easier procedure in which one could know the circumference of a circle if one could generate the square that has the same circumference now called the equal perimeter model. This was surprisingly simple to grasp and implement.

First of all, one can lay out a linear length, that divides by 4, lets say 28 which is 4 x 7. The length is made up of four lengths, each of 7 units and, a square of side length 7 will have a perimeter of 28, same as the linear length. The square is really just a rolled-up set of 4 lengths at right angles!

The diameter of a circle with 28 units on its circumference must be larger than its incircle of diameter 7 and, if pi is 22/7 then, the diameter will be exactly 14/11 of the side length. Notice that 14/11 is cancelling the seven and eleven in pi as 22/7.

The equal perimeter rope will be staked in the very center of the square. The side of 7 is then 7 x 14/11 or 98/11 units and this, times 22/7 equals 28 – the perimeter of both the circle, and square side-length 7 units. There is no need to calculate this if one draws a triangle ratio {11 14} from the center of the square. This triangle’s slope angle automatically “calculates” or reproportions the cardinal length (whatever this is) into a suitable rope (or radiant) length.

One often does not need to form the circle to know what its perimeter would be through measurement. Once one knows that every square has a twin circle of the same perimeter, this changes thinking. This is particularly significant when forming a circular model of the sun’s path in the year. If the “saturnian” year 364 days was used, it unusually divides by 28 days, and 13, and 7 days; the seven-day week. The square would have a side length of 13 weeks (91 days) and the radius rope would need to be (13 x 7) x 7/11 which, times 44/7 reconstitutes the circumference of 364 days.

My book Sacred Geometry: Language of the Angels has much to say on equal perimeter modelling, which is found throughout the ancient building traditions that followed on from the megalithic period, using the older techniques of metrological geometry alongside the development of arithmetic methods. Click on the Bookshop logo or Google, and find out more.

Organizing Ideas about Prehistory

For any activity to have a purpose there needs to be an organizing idea behind it and, in interpreting megalithic sites, many (often-competing) organizing ideas have been at work.

Archaeology has adopted different modus operandi over time, sometimes defining a new movement for the profession such as processual (New) and then post-processual (1980’s), and other specific ideas responding to changes in technology such as carbon dating or extrapolations of anthropological modalities expected for stone age peoples.

Ever since the rise of modern archaeology during the nineteenth century, the field has been in a perpetual identity crisis about its primary purpose. Archaeologists have never entirely agreed among themselves about what they are doing and why they are doing it. “What in fact is archaeology? I do not myself really know,” Mortimer Wheeler admitted in Archaeology from the Earth… [However] Most archaeologists would agree with one of Wheeler’s most eloquent statements: “The archaeologist is digging up, not things, but people.”

The Goddess and the Bull, Michael Balter, Free Press, 2005, p60
Continue reading “Organizing Ideas about Prehistory”