Number Proportion Anthropos- The Canon of Nineteen and the Classical Order

In ancient times the Pythagorean school and their latter followers of the Platonic and Neo-Platonic persuasions held that number had a character, beyond those of quantity and abstract mathematical operation.  In this aspect, each number is considered to represent symbolically a certain state of being.   For example, rather than being considered exclusively as a magnitude, the number one may be thought of as 'one-ness', the state of being unified;  two, as 'two-ness', the quality or state of division or separation;  three-ness as structure;  four-ness as materiality;  and so on.  It is this kind of number that is useful to the designer, not the quantitative or mathematical kind, which are used later in the building process.

A contemporary person needs to be cautious here as symbols are images that can have more than one meaning.  To utilize them in design we must in effect, be willing to enter a realm where 2 + 2 may not always and exactly equal 4.  If such ideas tend to cause a kind of intellectual nausea today we should recall that ancient consciousness on these subjects, and many others, was rather different from our own.  For example, ancient arithmetic had no decimals and could handle fractions only with great difficulty;  thus sums or products might be rounded off, so to speak, to the next higher or lower number.  Orders of magnitude, such as the difference between 216 and 2160, were not as clear-cut as they are for us, as the ancient arithmetic had no zero.  Beyond even these issues the ancients had no separate number 'sigils', such as 1,2,3, etc.  Instead they used small stones called 'calci', or they assigned numerical value to letters of the alphabet;  giving rise to the magical technology of 'gematria' in which words, phrases and names acquire numerical and even geometrical equivalents.  While these symbolic or 'fuzzy' aspects of ancient consciousness drive some modern 'hardhat' thinkers into histrionics there is no need for apprehension as long as we keep in mind that symbolic number lives in the human imagination while building construction must take place in the actual physical, measurable world;  and that there is no contradiction in this.

 In the Platonic tradition 'forms' in the imaginal realm are unchanging and timeless while their material representatives are in constant timely change.  To utilize symbolic number in design, or to bring any idea into actuality, we must ask ourselves 'how close an approximation is close enough?'  Is a ratio of 5/8 which equals .625, a fair representation of the Golden Section, or phi, Ø = .618…?  An aesthetic equivalency is here proposed if an interval differs from another by 5% of itself or less.  The intervals 5/8 and ? differ by .007… or about 1.3% of ?.  By this criteria then, the intervals are, for design purposes, equivalent.  We must keep in mind that design is not a branch of mathematics and we are not looking for the 'right' answer, but for an aesthetic effect that stimulates a kind of remembrance.

Although 'proportion' has a specific mathematical meaning - an equality between ratios, such as 2 is to 4 as 4 is to 8 [2:4 :: 4:8] - the term is taken here in a more general sense to indicate the use of number as a design tool.  Looking through the course of western history there seem to be three methods of application of symbolic number in design which may be called geometric, arithmetic and harmonic.  Geometric constructions are rather pure and maintain a close relation to unity but the intervals created are 'transcendental', they cannot be exactly expressed as whole numbers.  Arithmetic counts, such as we find in Vitruvius are clearly expressed but run the risk of loosing sight of unity.  Harmonic or small whole number ratios, as appear in the work of Palladio, strike a middle ground between the two other methods.  Another point of caution for the modern designer is that proportion and its methods are not to be mistaken for design itself.  A hammer is not an element of structure but it can be used to make a structure.  It is a means to an end.  But what, in design, is that end?  What kind of remembrance are we trying to stimulate?

Plato claims in the 'Republic' that sight is the most valuable perceptual sense because without it we would not be aware of the movements of the stars and planets, would not have learned how to count, and thus would not have achieved civilized status.  This is a dramatic assertion, implying that civilization itself is a design rooted in number.  This idea is a version of the later Hermetic soundbite 'As above, so below'.  In application this idea leads to the concept of Anthropos;  the analogy between the human being and the cosmos.  In this context civilization is a mezocosm, a middle realm between the individual, a microcosm, and nature, the macrocosm.  The mezocosm is a designed realm and its architecture, conceptual and physical is constructed with symbolic number.  In this theory the harmony created in architecture echoes and reinforces the social harmony sought by the human community in its public aspect, the polis or civitas.  It is this sense of unity that the designer ultimately seeks to stimulate.  It is in this context that the Roman philosopher Plotinus defines beauty as the 'memory of unity'.

The most obvious observable 'planetary' relationship is that between the sun and moon.  Though they 'meet' once a month at the new moon, because their cycles do not match exactly, the meeting takes place at a different position with regard to the fixed stars of the zodiac.  19 solar years of 365.25 days equals 6939 days while 235 lunar [synodic] months of 29.53 days also equals 6939 days.  Though named for the Greek astronomer Meton, this cycle was known to earlier civilizations.  Because the lunar 'nodes' cross the ecliptic with the same periodicity the arithmetic of eclipse prediction, the 'sine quo non' of magical governance in early civilization, became associated with the number 19.

The number 19 also plays a role in each of the four mathematical subjects of Plato;  the 'Quadrivium' of Arithmetic, Geometry, Music and Astronomy.  Arithmetically 19 may be taken as the sum of 7 and 12.  Both can be seen as combinations of 3 and 4, symbolizing the structuring of matter;  4 + 3=7 and 4 x 3=12 and 7:12 is a whole number approximation of Ø [  7/12 = .583…,  within 3.5% of ?].  In geometry the ratio of 18:19 is the relation of the altitude of the pentagon to its diagonal.  In music 19/18ths is one measure of the interval of the semitone [according to commentator Robert Lawlor].  In astronomy the number 7 is associated with the movements of the moon and 12 is associated with the sun.
The ancient Egyptians, from whom the Greeks got their higher education, had an aesthetic canon of 19 in which the height of the human body was divided into 19 units.  The number 19 may have been associated in the Egyptian mysteries with the magical animation of matter.  In figure 1 we see that in a figure divided into 19 parts, 4 parts give the position of the knees, 12 parts give the height of the upper legs and torso, the navel is at the Ø point, at the 11th unit above the ground [11/19 = .579, differing from ? by 3.9%] and the remaining 3 parts give the head.  This is also the formula for an archetypal classical architectural order, as given by Ware in the American Vignola, in which the pedestal is 4 units, the column is 12 units and the entablature is 3 units.

figure 1

Interestingly, the 18th unit on the figure marks the point on the forehead known in the east as the brow 'chakra'.  The Egyptian figure in the illustration wears a headband with a cobra projecting from his forehead at this point.  The serpent takes the same cyma recta shape and relative size as the topmost crowning moulding on the classical entablature.

In linking the numerical archetype of the classical order with the pre-western canon of 19 we see an anthropic analogy similar to that used by Vitruvius, in his well known 'Vitruvian Man' as drawn by DaVinci.  And Ø, the mystery factor that's not there in Vitruvius' canon in Book III, but is there within the arithmetic measures of the temple that he gives, emerges here to be a key factor.  Figure 2 shows the three proportional methods applied to the construction of an archetypal classical order of pedestal, column and entablature.  We take the overall height of the order to be 1 and divide the height into intervals of Ø and Ø2.  Arithmetically, Ø + Ø2 = 1 and because of the nature of Ø it is also true that Ø3 + Ø4 = Ø2;  thus Ø3 + Ø4 + Ø = 1.  Ø3 gives the pedestal, Ø gives the column height and Ø4 gives the entablature.  These geometrical increments vary less than 5% from the Ware's arithmetic canon of 19 and even less from the small whole number ratios 1/6, 5/8 and 1/5.  We may simplify the process further by dividing the overall height of the order into 5 parts.  The lowest part will be the pedestal.  Divide the remaining 4 parts into 5 smaller parts.  The top part will be the entablature.  The use of a proportional divider set to Ø will reduce this procedure to a few turns of the designer's hand.

figure 2

We may see here a kind of intersection of the three methods of application of proportion in architecture.  The existence of such intersections indicates that the three methods are not distinct and unrelated as some maintain but may be seen by designers as three ways to accomplish the same thing.  The anthropic analogy, the definition of beauty as the memory of unity and Plato's linking of beauty and the good provide hints of what this 'same thing' might be.

Steve Bass is an architect in practice in New York City since 1974.  He holds a Bachelor of Architecture degree from Pratt Institute;  a Master of Arts from the Royal College of Art, London, where he studied under the direction of Dr. Keith Critchlow;  and was a participant in the initial Prince of Wales's Summer Course in Architecture in 1990.  Mr. Bass is currently visiting Assistant Professor of Architecture at Notre Dame University for the fall '06-spring '07 year.  He is a Fellow of the Institute of Classical Architecture & Classical America where he has taught on the theoretical and applied aspects of proportion and geometry in design.  He has written for ICA&CA's journal 'The Classicist', for 'Traditional Building' magazine, and for 'American Arts Quarterly'.  His book, 'Proportion in Architecture' will be forthcoming from W W Norton in 2007-8.