Friday, January 28, 2011



The second known use of the magic square in Western art is the front piece to Arithmologia, a book authored by the great German Jesuit, Athanasius Kircher.

Kircher was a most unusual scholar of the seventeenth century; he was fluent in eleven languages including Latin, Hebrew, and ancient Egyptian.  He was a mathematician, a scientist, a geologist, an expert on Egyptian hieroglyphs, and was one of the first people to observe a micro-organism with a microscope.  

Kircher was also a prolific author having produced over forty works (including a book on China) during a period when official church censorship made it difficult to publish even one book.  Kircher had access to the great Vatican Library (perhaps old Chinese texts) and other church libraries that evaded other would be researches and authors.

The book Arithmologia is devoted to magic squares (the order 3 through the order 9 are demonstrated) as well as Pythagorean and other mathematical principles. 

Kircher and the Chinese

It seems plausible that Kircher was aware of the Chinese reverence for the Luo Shu as the 3x3 Magic Square is being held by a seraphin in Heaven as the model for the Pythagorean Theorem lay at Pythagoras’ feet on the terrestrial earth. 

Is Kircher demonstrating a connection with the Luo Shu and the Pythagorean Theorem? 

Although there is no proof that Pythagoras (or the Pythagoreans) knew of magic squares, the fact that a Pythagorean triplet of numbers occurs at the heart of any size magic square in the Luo Shu format and that these magic squares are excellent teachers of mathematics makes it plausible that Pythagoreans were fascinated with magic squares.  The Luo Shu was used in the Chinese civilization as early as one thousand years prior to the Pythagorean existence.

In addition, magic squares in the Luo Shu format fit Pythagorean numerology in the following ways:
  1. Odd numbers are superior to even numbers
  2. Powers of numbers play a significant role
  3. Pythagorean triplets exist in any order magic square in the Luo Shu format
  4. Both magic squares and Pythagorean numerology are great teachers of mathematics
  5. These magic squares serve as a model for Time and Space
Chinese mythology maintains the Luo Shu was a gift from Heaven to assist humankind to evolve and prosper.  Kircher seems to understand this cosmology of the Chinese not only by connecting the Luo Shu with the Pythagorean Theorem but also by having the Luo Shu appear in Heaven.

Kircher would often write in a manner that was intentionally not easy to comprehend in order for his books to pass scrutiny during the times of the Inquisition; the Index of Prohibited Books listed the banned books and an inquisition could be ordered for any person associated with these books.   Giordano Bruno burned at the stake and Galileo was placed under house arrest for their literary crimes and philosophies during this era.

Kircher attempted to reconcile this confusion with his art and the use of symbols in order to clarify the message that he was unable to articulate in his writings.

The early Chinese of four thousand years ago believed that Number, that is the Luo Shu, was presented to humankind from Heaven as this pattern of numbers appeared on the back of a tortoise that had crawled out of the Luo River.     The great Fu Hsi recognized the pattern of numbers as sacred and soon temples and cities were being designed based on the model of the Luo Shu.

Pythagorean Symbolism

The art piece above demonstrates classic Pythagorean symbolism:
  • The tetractys
  • the proof of the Pythagorean Theorem
  • The Luo Shu or 3x3 Magic Square
  • Heaven and the terrestrial earth
  • Numerology
  • The enneagram
  • The planetary spheres
  • The pentagram and hexagram
Pythagorean Symbolism from the Royal Tombs of China

The legendary sage Fu Hsi and Nu Wa holding their customary symbols, the carpenter’s square (ju) and the compass (gui).  These symbols represented the yang / yin balance of male and female, Heaven and earth, odd and even.
The carpenter’s square also symbolizes humankind’s need for math, numbers, and measurement and in the Chinese language can be synonymous with the gnomon (ju). [gnomon. Gr., can mean interpreter, one who knows, rule of faith or conduct; canon or tenet]

The Compass and Square

The combination of the carpenter’s square and compass form a new word, gui ju and projects the important concept of establishing order (over chaos), the moral standard or the way things should be.   This is an example of how the early Chinese incorporated mathematical philosophy into the structure of their language.

The Luo Shu was considered a model for Time and Space and had functional use to humankind.  Together with the gnomon, the Luo Shu was a language gifted from above that helped to connect Heaven and earth via the observation of the Sun (rather the shadow of the gnomon cast by the Sun) and the understanding of math. 

Is this the message of Athanasius Kircher’s Arithmologia?

Saturday, January 22, 2011



The first known use of magic squares in Western art was the 4x4 magic square in Albrecht Durer’s Melencolia I (1514).  

This was during the embryonic phase of the Italian Renaissance with legends like Brammante, Da Vinci, and Pacioli introducing math into art and architecture – an ancient concept which was enjoying a rejuvenation of sorts.*

Plato was an important influence to the philosophies of the great mathematicians (artists and architects) during the Italian Renaissance as his books were just coming onto the market after being out of print for about a thousand years (the "Dark Ages").

At the same time, books on alchemy and magic were introducing magic squares to the public.  Luca Pacioli and Cornelius Agrippa would publish books with a set of magic squares and their magical correspondences.  

The symbolic significance of the 4x4 magic square corresponded to related Pythagorean themes:
  1. Plato’s mantra that “number” was essential for the evolution or destiny of humankind as Number could provide the way to measure.

  2. An alchemical or "magical" correspondence to the planet Jupiter; that is, part of a complicated system where powers of numbers related to magic squares and a mathematical language that could explain the universe.  
The 4x4 magic square corresponded to Jupiter and could be used to invoke the spirits of Jupiter to counter the effects of Saturn, that is whatever affect Saturn has on the planet earth and its inhabitants - which in this case could be the psychological affects of the mind and its physical  impact on the body.  What was referred to as "Melancholia" in the sixteenth century may be commonly known today as Major Depression, which can be associated with chronic widespread pain and Fibromyalgia.

Plato, from the Republic

“And haven’t measuring, counting, and weighing come to light as the most charming helpers in these cases?  As a result of them, we are not ruled by a thing’s looking bigger or smaller or more or heavier; rather we are ruled by that which was calculated, measured, or, if you please, weighed.”

Other Platonic (measuring) symbols in Durer’s Melencolia I include:
  • The carpenter’s T-square
  •  The compass
  •  The ruler
  • The hourglass
  • The scales
Additional symbols include a bell, a pot of gold, keys, a purse, a ladder with seven rungs, a polyhedron, a brooding Goddess with her muse, a brooding dog, carpenter’s tools and materials, and more.

There are 880 different ways to arrange the numbers 1 thru 16 in a 4x4 magic square, Durer chose the one square that appeared in Fra. Luca  Pacioli's book De Viribus Quantitatis (1509) with these particulars:.
  • The numbers fourteen and fifteen are arranged to identify the date of this masterpiece:  1514.
  • Each row, square, and major diagonal add up to 34 (the conventional definition of a magic square)
  • Each quadrant of four numbers of the square add up to 34
  • The middle four numbers add up to 34
  •  The numbers in the four corners add up to 34
  • The middle two numbers of the first and last row add up to 34
  •  The middle two numbers of the first and last column add up to 34
The use of magic squares in art was a reference to a Platonic or Pythagorean world view that emphasized number as the most integral part of human existence.**

This paralleled the ancient Chinese reverence for number (the Luo Shu) as represented in their art (the TLV Bronze Mirror, Jade Bi disc) and architecture and should be considered as an influence on Platonic, Neoplatonic, Pythagorean, and early Christian philosophy and numerology.  

*  The magic square was part of an elaborate "sacred geometry" that was incorporated into the design of metal and jeweled covers of Illuminated Manuscripts during the Carolingian era (example: upper cover of the Lindau Gospels, c. 975 AD).  Monks from monastaries (such as Hrabanus Maurus) incorporated math (numerology and magic squares) into poetry, art, and the design of several bookcovers during this period.

**  The use of  magic squares in art can also make alchemical correspondences to the planetary spheres that was part of a complicated system connected to Renaissance magic.