Hertha Ayrton remarked that “the Doric architect worked within a peculiarly rigid framework. He was not required to innovate on a large scale and would not have been admired had he done so, or it would have meant that he was not fulfilling the task he had been set, but performing a different one.” Her statement adheres remarkably with the way Classical period structures follow the proportional formula based on the square roots of primary numbers from 4 – 8. Subsequently, type 6-11 temples have a value equal to 2, or the square root of four. Temples that are type 6-12 follow a proportional rule equal to the square root of 5 (2.2361); type 6-13 temples have a value equal to the square root of 6 (2.4456); type 6-14 temples have a ratio equal to the square root of 7 (2.6458); and type 6-15 temples have a value equal to the square root of 8 (2.8284).
In practical terms, the length of the temple of Zeus at Olympia, for example, can be accurately calculated by multiplying the means product, or 2.4456, by the sum of the structures interaxial using the regression coefficient formula ( Σx:Σy = 2.4456), thus, the temple width (25.265m) times 2.4456 gets a result of 61.788m. This compares with the known interaxial length for the temple of Zeus of 61.706m; an error of only 0.13%.
To know the degree or strength of the relationship that exists between the two variables, an 'r' coefficient (Pearson product-moment correlation) is used to measure the degree each data point deviates from the regression least-square line. The numeric value is measured from –1 to +1. The closer the data pints are to the regression line, the grater the magnitude of 'r' and the closer the value will be to +1. As the value of 'r' approaches -1, the relationship between data points and the regression line grows weaker, as does the potential for establishing reliable predictions. The formula for determining 'r' (correlation) is:
In the example of the temple of Zeus at Olympia, the correlation between width and length is significant. To quantify that significance, the r2 value for that temple is an astounding 99.89%, or we can predict either the width or length of the temple with a degree of certainty equivalent to 99.89%.
The analysis of the statistical tests indicates that temple relationships are based on exponential squares rather than linear modules -- a proportional formula not based on an arithmetic straight-line relationship, but rather an exponential curve of irrational numbers. To illustrate how this looks, examine Table 3. Here both Tobin’s Archaic rule and Coulton’s Sicilian rule are compared with the non-linear curve of whole number square roots.
Vitruvius, certainly helps to validate that some secrets were best kept concealed. As Pappus (300 ad.) recorded:
That is exactly what legend said became of Hippasus for revealing the construction of the pentagram; which is tied directly to the discovery of irrational numbers. Possibly religious architectural design was maintained within Pythagoreans’ secret society. It was Pythagoras who founded the brotherhood of mathematicians on which the modern Masonic order is based. W. L. Wilmshurst writes in The Meaning of Masonry [1980:26] that it is “…perfectly certain that Pythagoras was a very highly advanced master in the knowledge of the secret schools of the Mysteries, whose doctrine…is enshrined for us in our Masonic system.”
The Pythagorean Order was established in Sicily and Calabra about 500 B.C. and “…consisted of three degrees: the novices or ‘politics’ (esoteric state), the ‘monothets’ (first degree of initiation), active philosophers who directed the social and political work of the Society by instructions given to the novices, and lastly, the ‘Mathenaticians’ who had been through the complete initiation, including of course the ‘Laws of Numbers’
[Ghyka 1977:115, n2]
In the Masonic order every Brother admitted into its ranks passes though successive stages until they are initiated into the Greater Mysteries. Likewise, in the Pythagorean School, adherence to an architectural uniformity without deviation, points to some form of exclusive knowledge being transmitted through the order of master masons and architects. As Dallaway states:
We may conclude that the craft or mystery of architects and operative masons was involved in secrecy, by which knowledge of their practice was carefully excluded from the acquisition of all who were not enrolled in their fraternity.
[quoted by Oliver 1975:6]
Wilmshurst elaborates on this perspective:
It is difficult to assimilate all architects of the 4th and 5th centuries as disciples of the so called 'Mystic of Numbers' theory founded by Pythagoras, but we can admit that this philosophical and mathematical doctrine seduced a certain number of followers who conformed to a tradition by achitects for them to retain all knowledge of their academic careers. Therefore, there was a veritable 'bulders secret' to conserve a virtual monopoly on construction methods and knowledge.
With no concrete information on ancient architecture available, we have no choice but to formulate hypothetical propsitions. Our opinion is that the masters of Greek architecture used the mystical, or so called divine power of numbers, as an excuse to preserve the secrecy of their trade, i.e...the chosen few. [Adams 1973:222]
It is this veiled illusion to a mathematical codex cryptically concealed in the Masonic Brotherhood that may have been guarded from Vitruvius. The Masonic guild records within its statutory code an obligation for the Mason not to betray the secrets of his craft (Cooke Manuscript in the British Museum, copy dating from ca. 1430 of a fourteenth century text). In another, somewhat later, Masonic manuscript, we find the admonition “You shall keepe secret ye obscure and intricate pts, of ye science, not disclosing them to any but such as study and use the same [MSS n=2, Breat Lodge of London].
Once we make the transition between irrational numbers, proportion theory and temple architecture, the element of Pythagorean fraternity must be part of the formal solution. Malcolm Bell [1980:368] says that the coincidence of Pythagorean numerology and the Temple of Olympia at Akragas is “especially significant.” What Bell was referring to was the stylobate width/length ratio of 27:13. The reference is to “melodic interval” directly taken from the Pythagorean Philolaos, between the whole musical tone and the smaller “semitone” [1981:369]. Not only this is the difference a harmonic interval, but also the ratio of 27:13 is an irrational number.
Sir Thomas Heath [1981:85, n2] quotes a fragment of Archytas’s work, Of Music, and defines harmony as when “…by whatever part of itself the first exceeds the second, the second exceeds the third.” If we examine this concept in relation to Doric temples there seems to be a harmonic correlation between type styles. This harmonic relationship is shown in Figure B. The origin of harmonic proportion was, according to Heath [1981:86], “…discovered by the Babylonians and first introduced into Greece by Pythagoras.” It was used, Imblichus says, by many Pythagoreans. Imblichus seems to know what he’s talking about and further defines the Pythagoreans by saying they have “…devoted themselves to mathematics…and seeing [equally] the facts about harmony…so that whoever wishes to comprehend the true nature of existing things should turn his attention to these, that is to numbers and proportions, because it is by them that everything is made clear.”
We can conclued, the results produced a mathematical model that is better than 99% accurate in predicting Classical-period Doric temple Width:Length proportions. Not only do the statistical results produce a mathematically reliable model, but also imply that the progenitor of the formula was Pythagoras since the results produced a proportional formula based on irrational numbers – a mathematical phenomena first revealed within the Pythagorean brotherhood. This conclusion is reached since Doric temple proportionality crystallized around 490 B.C., and remained virtually unaltered throughout the tenure of the Doric style.
The science (or knowledge) had its origin in the sect (or school) of Pythagoras…[and] was so affected by its reverence for things that a saying became current in it, namely, that he who first disclosed the knowledge of surds or irrationals and spread it abroad among the common would perish by drowning.
[translation by Burkert 1972:457]
A Master Mason, then, in the full sense of the term, is no longer an ordinary man, but a divinized man, one in whom the Universal and the personal consciousness have come into union. Many of the Euclidean and Pythagorean theorems, now regarded merely as mathematical demonstrations, were originally expressions, veiled in mathematical glyphs, of the esoteric science of soul-building or true Masonry. The well-known 47th Proposition of the First Book of Euclid is an example of this and in consequence has come (though few modern Masons could explain why) to be inscribed upon the Past Master’s official jewel. Again, the squaring of the circle – that problem which has baffled so many modern mathematicians – is an occult expression…which has come into the present Masonic rituals from Pythagorean Geometry and pagan religion.
The following conclusions can be drawn from the statistical results:
There was a break in the proportional system that took place between the Archaic and Classical Periods (535 B.C. and 490 BC).
The use of irrational numbered formed the basis for Classical Doric temple proportions.
The use of irrational numbers for designing Doric temples is rooted in theoretical geometry rather than simple arithmetic means.
The Pythagorean School of architects likely provided the overall design for Doric temples, especially regarding their proportions.
But does the formula work? A Look at Practical Applications
The proof to any theory is whether it works in practice. To test the theory I have used the temple of Poseidon at Isthmia and also the Artemis temple at Kalydon.
At Isthmia, the temple floor plan was reconstructed as part of excavations conducted in 1989. Currently, little remains of the structure other than the stylobate. However, with a degree of accuracy the column height can be descerned by taking a statistical sample of all 6 13 temples and compairing column heights, then compute the statistical probability for the Isthmia temple. Using that method, I calculated this temple's column height at 8.493 meters. This technique works well for temples with 13 columns on the long side, but does not hold up as well for temples with 12 and 14 columns, since there is not large enough sample to work from.
Nothing remains of the Artemis temple other than the stylobate and crepes. The remains of the temple indicate it was constructed about 360 B.C. Three other temple building periods have been detected, each oriented north-south with the entrance on the south. Although a number of measurements have been taken, the column height was not one of them. Using the same statistical model as before, I can calculated the column height to be @ 4.828 meters.
Another example to test the statistical method is that of the temple of Athena at Karthaia (Keos) described by Woodward (2012). Although he does not show the corner contractions, the result of the width length ratio is 2.009, exactly that of the square root of 4, which it should be for temples with 6 x 11 columns. With this information we can calculate the column height to be about 5.103 meters. More recent photos of the site show a column reconstructed, but I have not found any information on its dimensions.