Proportion and Analogy
“6.522 Es gibt allerdings Unaussprechliches. Dies zeigtsich, es ist das Mystische (There is indeed the inexpressible, it shows itself. It is what is mystical).”
Ludwig Wittgenstein, 1921.
The idea of measurement preceded the idea of number.
In order to measure something, it must be related to something else.
All that is required for a comparison to be made (or, that is, for a measurement to be taken) is that the thing being measured shares a quality in common, with the thing being used as a rule. In other words, to measure one thing is to express it in terms of something else. In symbolic comparison, we are familiar with the Latin word, ‘ratio,’ but the Greeks named such a statement-of-measure λόγος (logos).
The observation of qualitative alikeness is surely among the most rudimentary functions of all complex mentality.
Common types of measure—including length, mass, volume, or size—were named μέρος, or ‘magnitude.’ To the Greek mind, magnitude and number were distinguished from one another.
Of a shared quality, take for example two lines of arbitrary length. They can be compared in terms of their length. That they are equal or unequal might be observed, as well as which is longer and which is shorter. These are qualitative measurements, and are inherent to the form of the objects in question. More complex geometric operations can also be performed, which amount to the four basic functions of arithmetic, all entirely without number. Two magnitudes may be appended to produce a sum, one may be removed from another to yield a difference. Two linear magnitudes can be multiplied to form a plane the area of which is their product, and division—that is, the severance of equal components from a whole—can likewise be constructed geometrically, although the process is rather involved and would be overly laborious to write out here.
A λόγος, (or ratio) does two things. It establishes a fundamental standard, the consequent, and it expresses something in terms of that standard, the antecedent. That is the simple mystical operation underlying all proportion.
The creation of terms of expression has long been a profound, and pervasive occult principle.
The Pythagoreans contended that only if whole, equal components of the antecedent can be measured by the consequent, does the ratio of magnitudes constitute a number. The given magnitudes are then said to be commensurate or, co-measurable. Magnitudes which cannot be expressed as a number in this way are incommensurate. Though an incommensurate thing cannot be expressed as a number, it can be expressed as a single, resultant magnitude. If one magnitude, for example is the length of the side of a square, and the other is the length of its diagonal, then the magnitude resultant from their ratio cannot be fit to a number. The given magnitudes are incommensurate. The thing exists, but cannot be put to strict language; it can only be demonstrated.
While two magnitudes of different quality or type cannot be put to a ratio with one-another, the geometer Eudoxis described a theory of proportion that permitted the comparison of ratios themselves—because commensurate ratios do constitute numbers. For example, a cone and a cylinder of the same base and height will stand in proportion such that the ratio between their volumes, and the ratio 1:3, are themselves equivalent. Which is to say, that proportion is a constant relationship between ratios, no matter the quality. Proportion, then, is understood to bely a divine governing language, through which we can glimpse the metaphysical connection between seemingly disparate things.
Man of antiquity used his body as the standard of measurement for all things. The simplest cubit was defined as the magnitude from the elbow to the tip of the fingers. The Greeks observed of their body that four dactyls comprised the width of a palm. A foot was divided as well into dactyls. Larger units of measure were configured from these basic units; a stadion being six-hundred feet, and five-thousand being a mile. Of course there was somewhat little in the way of standardization. It’s well known that—for example—athletes competing in a footrace in one place, may be running on a slightly longer track than another place, as a foot here was slightly longer than a foot there. Standardization of units themselves is not so important to the discussion of proportion.
That one ‘rule’ would be standardized is a materialistic necessity. It serves the tax collector, and the merchant. Conversely, consider that any human body—no matter its individual size relative to another—contains within itself the same rules as any other. That the body expresses certain proportional relationships, in terms of its own λόγος, is always the same. These rules are true, local to any human system. So we understand that a standardized rule is a mundane imposition upon the world, by man. λόγος, by contrast, is a divine formula that characterizes the natural world, and describes the relationships between its constituent parts. As Vitruvius says,
“…the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; the head from the chin to the crown is an eighth, and with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth; from the middle of the breast to the summit of the crown is a fourth. … The other members, too, have their own symmetrical proportions, and it was by employing them that the famous painters and sculptors of antiquity attained to great and endless renown.”
Anaxagoras recognized the radical, mystical importance of measurement. He wrote that “The purpose of life is the investigation of the Sun, Moon, and the Heavens.”
By virtue of its form, the symbol contains within itself mystical imperatives, which are released to posterity despite intervening generations of intellectual ignorance. Eliade described the resilience of the symbolic arts in this peculiar respect. He wrote, “These degraded images present to us the only possible point of departure for the spiritual renewal of modern man. … Modern man is free to despise mythologies and theologies, but that will not prevent his continuing to feed upon decayed myths and degraded images.” The wealth of the traditional, qualitative mentality, he explains, lies “…hidden. The whole treasury of myths, ‘laicized,’ and ‘modernized’.”
Proportion is this: Correspondence between the magnitudes of individual parts of a work, and, the correspondence between the entire work and a certain part set out as the rule.
Perhaps the most critical proportion evident in the human body, is the squared circle. Vitruvius explains:
“Then again, in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centered at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square.”
To attenuate the language a bit, we may say that the body contains within itself a proportional relationship between the square and the circle. Any student of the Hermetic arts will no doubt understand the significance of the central symbol of the alchemical opus—the transfiguration of the self—being contained within the proportional system of the human body.
To apprehend the Greek idea of proportion as it came to define an architectural canon, we begin with the word itself. Like ratio, proportion is loaned to modern English speakers from Latin. Recall that a ratio was known to the Greeks simply as ‘λόγος’ (logos). The word for what we call ‘proportion’ in architecture, was ‘ἀναλογία’ (analogia). We are confronted with the fact that the standard of comparison in sacred geometry, the λόγος, formed the basis of the name given to the proportional system governing the temple: Ana-logos.
λόγος descends from the same Proto-Indo-European root as the Latin word “legio,” which is where the modern English word ‘legion’ originates, but which literally meant ‘chosen’ in the Roman tongue; in the military context, something like ‘chosen warriors.’ The Proto-Indo-European root likewise most nearly corresponds to something like ‘to gather up,’ ‘to choose,’ or ‘to collect.’ Heraclitus was the first to apply λόγος as a philosophical principle—signifying those words which are ‘chosen,’ and arranged to physically convey meaning.
‘ἀνά’ (ana-) is typically be translated to ‘up.’ However, ἀνά can also translate more nearly to something like the English prefix ‘re-’, as in reuse, remake, or regard. Therefore, ἀνά might be likened to ‘again.’ In the case of ἀναλογία, it is this second sense from which I believe the word’s original, literal meaning was formed, and lent to the matter of architectural proportion by the Greek masters.
In the sense of measurement, ἀναλογία might thus be translated as ‘[to take] the standard (λόγος), again.’ Therefore ἀναλογία refers to the process of deriving what-comes-after, given a rule, from what-comes-before. Further then, it corresponds to the reflexive quality of a thing which expresses by virtue of its form, the standard(s) which gives it form.
In the schools today proportion is typically presented in passing, as an aesthetic novelty. It’s something almost frivolous, which is perpetrated upon the architecture according to the superstitious tastes of men who were yet-to-develop a more mature, industrious mentality. Proportion, to the academic, is a matter of taste, fashion, or style.
The man who would, in antiquity, have been a priest to a degenerate orthodoxy is today a committee-member, a bureaucrat, a politician’s assistant, or an academic at some university. Yesterday he played with his scales and cuttings of coins, today he fiddles with his lists and his tables just the same. He shuffles things around to reflect the will of the funding, and jockeys for the approval of the fellows who feed him and who keep him fat. So it is with the intellectuals.
That beauty, in the Vitruvian sense of the word, is defined in terms of proportion is known to intellectuals. They are aware of it. The intellectual may realize that proportion was a thing of some importance, but not how or why. Of proportion he sees only its mundane, aesthetic aspects. Those over-crude and secularized items, which simply look a certain way, are all that he means when his mouth makes the shape of the word. For the intellectual, the word is only a category.
The Greeks did not employ drawing in the architectural process, to any serious degree. They conducted their architecture in accordance with a canon. The canon was gathered up, upon a foundation of universal principles. The mechanism of that foundation was proportion. Given several parameters (λόγος), the temple was designed analogically, in-situ, throughout the construction process. Any given state of the system defined the subsequent states.
That is what we mean, when we say that proportion is a mystical methodology, not a style. The aesthetic of the proportional method is, itself, incidental. Mistaking a manner of making, with a manner of looking, is one of the most garish symptoms of modernity.
The Greek architects did not draw because it would have been prohibitively difficult to laminate enough individual pieces of parchment, or papyrus, to form a writing surface large enough to work to any real degree of detail. Therefore it fell to proportional relationships to lend a measure of predictability to the design process. In order to ensure that the relation between constituent parts remained proportionate, the architect supplied several tools. The exact nature of these tools is not clearly known; we know their names, and their general function thus-far described, but not their particular form. We know that a verbal description of the structure’s dimension and general configuration was common. From that building account, enough was agreed upon to begin construction according to the canon.
Two tools, peculiar to the architects trade, were called ‘ἀναγραφεύς’ (anagrapheus) not to be confused with the Byzantine bureaucratic office of the same name, and ‘παραδειγμα’ (paradeigma). Broadly speaking, both were standards of a sort—against which individual components of the temple could be checked and refined by workers and tradesmen, on-site, during construction. It is for this reason that the compass, or calipers, are often depicted as the tools of the maker of the universe. The compass allows for a workman to transfer measurements from a divine standard onto a mundane system. They are what allows him to create his temple proportionate to, and therefore in the image of, the world.
Bearing in mind the philosophical implications of proportion to the Greeks, consider the following passage from the fifth book of The Histories of Herodotus, where it is said that the architects were able to produce the Temple of Delphi to such a quality as to surpass the paradigm itself:
“After fortifying Lipsydrium north of Paeonia, they [the Athenians], in their desire to use all devices against the sons of Pisistratus, hired themselves to the Amphictyons for the building of the temple at Delphi which exists now but was not there yet then. Since they were wealthy and like their fathers men of reputation, they made the temple more beautiful than the model [παραδειγμα] showed. In particular, whereas they had agreed to build the temple of tufa, they made its front of Parian marble.”
There existed for the ancients a continuum between body, temple, and cosmos. Through his body—and the physical intellect—man learns to measure. He is endowed with purpose. By measurement, man observes the heavens. In observing the heavens, man divines and records (καταλέγω) its governing principles. Those principles form a set of proportions. With proportion, man constructs his temple in the image of the world, expressed holistically in terms of his body.
Sacred architecture thus acquires a reflexive character. The temple is a model of the cosmos, which is derived through adherence to the natural λόγος. It expresses, by nature of its form and construction, the specific rules and relations which govern it, in the same way that the cosmos announces to man its own governing principles, in terms of him.
On precisely this reflexivity, Agrippa wrote:
“No one can excel in the alchemical art without knowing the principles in himself, and the greater the knowledge of the self, the greater will be the magnetic power attained thereby and the greater the wonder’s to be realized.”