Da Vinci’s First Code: Our Numbers Our Gods, the Pentagram
with illustrated examples ranging from da Vinci's three portraits of women to the Great Pyramid and the Parthenon.

Renaissance neoplatonist speculators used the Golden means, or ‘Ratio’ for  the pentagon and the pentagram necessary for the construction of the of the Platonic solids.

But irrespective of the base that any of the earlier cultures or even civilizations chose, the first group of numbers to be appreciated and understood at some level was the group of whole numbers (or natural numbers). These are the familiar 1, 2, 3, 4.... Once humans absorbed the comprehension of these numbers as abstract quantities into their consciousness, it did not take them long to start to attribute special properties to numbers. From Greece to India, numbers were accredited with secret qualities and powers. Some ancient Indian texts claim that numbers are almost divine, or "Brahma-natured.

These manuscripts contain phrases that are nothing short of worship to numbers (like "hail to one"). Similarly, a famous dictum of the Greek mathematician Pythagoras (whose life and work will be described later in this chapter) suggests that "everything is arranged according to number." These sentiments led on one hand to important developments in number theory but, on the other, to the development of numerology-the set of doctrines according to which all aspects of the universe are associated with numbers and their idiosyncrasies.

To the numerologist, numbers were fundamental realities, drawing symbolic meanings from the relation between the heavens and human activities. Furthermore, essentially no number that was mentioned in the holy writings was ever treated as irrelevant. Some forms of numerology affected entire nations. For example, in the year 1240 Christians and Jews in Western Europe expected the arrival of some messianic king from the East, because it so happened that the year 1240 in the Christian calendar corresponded to the year 5000 in the Jewish calendar. Before we dismiss these sentiments as romantic naivete that could have happened only many centuries ago, we should recall the extravagant hoopla surrounding the ending of the last millennium.

One special version of numerology is the Jewish Gematria (possibly based on "geometrical number" in Greek), or its Muslim and Greek analogues, known as Khisab al Jumal ("calculating the total"), and Isopsephy (from the Greek "isos," equal, and "psephizein," to count), respectively. In these systems, numbers are assigned to each letter of the alphabet of a language (usually Hebrew, Greek, Arabic, or Latin). By adding together the values of the constituent letters, numbers are then associated with words or even entire phrases. Gematria was especially popular in the system of Jewish mysticism practiced mainly from the thirteenth to the eighteenth century known as cabala. Hebrew scholars sometimes used to amaze listeners by calling out a series of apparently random numbers for some ten minutes and then repeating the series without an error. This feat was accomplished simply by translating some passage of the Hebrew scriptures into the language of Gematria.

One of the most famous examples of numerology is associated with 666, the "number of the Beast." The "Beast" has been identified as the Antichrist. The text in the Book of Revelations (13:18) reads: "This calls for wisdom: let anyone with understanding calculate the number of the beast, for it is the number of a man. Its number is six hundred and sixty-six." The phrase "it is the number of a man" prompted many of the Christian mystics to attempt to identify historical figures whose names in Gematria or Isopsephy give the value 666. These searches led to, among others, names like those of Nero Caesar and the emperor Diocletian, both of whom persecuted Christians. In Hebrew, Nero Caesar was written as (from right to left): xqw off, and the numerical values assigned in Gematria to the Hebrew letters (from right to left)-50, 200, 6, 50; 100, 60, 200-add up to 666. Similarly, when only the letters that are also Roman numerals (D, I, C, L, V) are counted in the Latin name of Emperor Diocletian, DIOCLES AVGVSTVS, they also add up to 666 (500 + 1 + 100 + 50 + 5 + 5 + 5). Clearly, all of these associations are not only fanciful but also rather contrived (e.g., the spelling in Hebrew of the word Caesar actually omits a letter, of value 10, from the more common spelling).

Amusingly, in 1994, a relation was "discovered" (and appeared in the Journal of Recreational Mathematics) even between the "number of the Beast" and the Golden Ratio. With a scientific pocket calculator, you can use the trigonometric functions sine and cosine to calculate the value of the _expression [sin 666° + cos (6 x 6 x 6)°}. Simply enter 666 and hit the [sin} button and save that number, then enter 216 (= 6 x 6 X 6) and hit the [cos} button, and add the number you get to the number you saved. The number you will obtain is a good approximation of the negative of phi. Incidentally, President Ronald Reagan and Nancy Reagan changed their address in California from 666 St. Cloud Road to 668 to avoid the number 666, and 666 was also the combination to the mysterious briefcase in Quentin Tarantino's movie Pulp Fiction.

One clear source of the mystical attitude toward whole numbers was the manifestation of such numbers in human and animal bodies and in the cosmos, as perceived by the early cultures. Not only do humans have the number 2 exhibited all over their bodies (eyes, hands, nostrils, feet, ears, etc.), but there are also two genders, there is the Sun-Moon system, and so on. Furthermore, our subjective time is divided into three tenses (past, present, future), and, due to the fact that Earth's rotation axis remains more or less pointed in the same direction (roughly toward the North Star, Polaris, although small variations do exist), the year is divided into four seasons. The seasons simply reflect the fact that the orientation of Earth's axis relative to the Sun changes over the course of the year. The more directly a part of the Earth is exposed to sunlight, the longer the daylight hours and the warmer the temperature. In general, numbers acted in many circumstances as the mediators between cosmic phenomena and human everyday life. For example, the names of the seven days of the week were based on the names of the celestial objects originally considered to be planets: the Sun, the Moon, Mars, Mercury, Jupiter, Venus, and Saturn.

The whole numbers themselves are divided into odd and even, and nobody did more to emphasize the differences between the odd and even numbers, and to ascribe a whole menagerie of properties to these differences, than the Pythagoreans. In particular, we shall see that we can identify the Pythagorean fascination with the number 5 and their admiration for the five-pointed star as providing the initial impetus for the interest in the Golden Ratio.

Author Roger Herz­Fischler traced the fallacy of the Golden Ratio as Pacioli's canon for proportion to a false statement made in the 1799 edition of Histoire de Mathematiques (History of mathematics) by the French mathematicians Jean Etienne Montucla and Jerome de Lalande.

The third volume of the Divina (A short book divided into three partial tracts on the five regular bodies) is essentially an Italian word­by-word translation of Piero's Five Regular Solids composed in Latin. The fact that nowhere in the text does Pacioli acknowledge that he was merely the translator of the book provoked a violent denunciation from art historian Giorgio Vasari. Vasari writes about Piero della Francesca that he

... was regarded as a great master of the problems of regular solids, both arithmetical and geometrical, but he was prevented by the blindness that overtook him in his old age, and then by death, from making known his brilliant researches and the many books he had written. The man who should have done his utmost to enhance Piero's reputation and fame, since Piero taught him all he knew, shamefully and wickedly tried to obliterate his teacher's name and to usurp for himself the honor which belonged entirely to Piero; for he published under his own name, which was Fra Luca dal Borgo (Pacioli), all the researches done by that admirable old man, who was a great painter as well as an expert in the sciences.

So, was Pacioli a plagiarist? Quite possibly, although in Summa he did render homage to Piero, whom he regarded as "the monarch of our times in painting" and one who "is familiar to you in that copious work which he composed on the art of painting and on the force of the line in perspective.

R. Emmett Taylor (1889-1956) published in 1942 a book entitled No Royal Road., Luca Pacioli and His Times. In this book, Taylor adopts a very sympathetic attitude toward Pacioli, and he argues that, on the ba­sis of style, Pacioli may have had nothing to do with the third book of the Divina and it was just appended to Pacioli's work.

Be that as it may, there is no question that if not for Pacioli's printed books, Piero's ideas and mathematical constructions (which were not published in printed form) would not have reached the wide circulation that they eventually achieved. Furthermore, up until Pacioli's time, the Golden Ratio had been known only by rather intimidating names, such as "extreme and mean ratio" or "proportion having a mean and two extremes," and the concept itself was familiar only to mathematicians.

The publication of The Divine Proportion in 1509 gave a new topical interest to the Golden means. The concept could now be considered with fresh attention, because its publication in book form identified it as worthy of respect. The infusion of theological/philosophical meaning into the name ("Divine Proportion") also singled out the Golden Ratio as a mathematical topic into which an increasingly eclectic group of intellectuals could delve. Finally, with Pacioli's book, the Golden means started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use.

Leonardo da Vinci's drawings of polyhedra for The Divine Proportion. drawn (in Pacioli's words) with his "ineffable left hand," had their own impact. These were probably the first illustrations of skeletal solids, which allowed for an easy visual distinction between front and back. Leonardo may have drawn the polyhedra from a series of wooden models, since records of the Council Hall in Florence indicate that a set of Pacioli's wooden models was purchased by the city for public display.

In addition to the diagrams for Pacioli's book, we can find sketches of many solids scattered throughout Leonardo's notebooks. In one place he presents an approximate geometrical construction of the pentagon. This fusion of art and mathematics reaches its climax in Leonardo's Trattato della pittura (Treatise on painting; organized by Francesco Melzi, who inherited Leonardo's manuscripts), which opens with the admonition: "Let no one who is not a mathematician read my works"-hardly a likely statement to be found in any contemporary art handbook.

The drawings of solids in the Divina have also inspired some of the intarsia constructed by Fra Giovanni da Verona around 1520. Intarsia represent a special art form, in which elaborate flat mosaics are constructed of pieces of inlaid wood. Fra Giovanni's intarsia panels include an icosahedron, which almost certainly used Leonardo's skeletal drawing as a template.

The lives of Leonardo and Pacioli continued to be somewhat intertwined even after the completion of The Divine Proportion. In October of 1499 the two men fled Milan, after the French army, led by King Louis XII, captured that city. After spending brief periods of time in Mantua and Venice, both settled for some time in Florence.

During the period of their friendship, Pacioli's name became associated with two other major mathematical works-a translation into Latin of Euclid's Elements and an unpublished work on recreational mathematics. Pacioli's translation of the Elements was an annotated version, based on an earlier translation by Campanus of Novara (1220-1296), which appeared in printed form in Venice in 1482 (and which was the first printed version). Pacioli did not manage to publish his compilation of problems in recreational mathematics and proverbs De Viribus Quantitatis (The powers of numbers) before his death in 1517. This work was a collaborative project between Pacioli and Leonardo, and Leonardo's own notes contain many of the problems in De Viribus.

Fra Luca Pacioli certainly cannot be remembered for originality, but his influence on the development of mathematics in general, and on the history of the Golden Ratio in particular, cannot be denied.

But the documented history of the Golden Ratio is inconsistent with the idea that this proportion was particularly revered by artists in the centuries preceding the publication date of Pacioli's book. Furthermore, all the serious studies of the works of the three artists by art experts (e.g., Giotto by Francesca Flores D'Arcais; Cinzabue by Luciano Bellosi) give absolutely no indication whatsoever that these painters might have used the Golden Ratio-the latter claim appears only in the writings of Golden Number enthusiasts and is based solely on the dubious evidence of measured dimensions.

And some authors even attribute the invention of the name "the Divine Proportion" to Leonardo da Vinci himself. The discussion usually concentrates on  the unfinished canvas of "St. Jerome," the two versions of "Madonna of the Rocks,"and  the drawing of "a head of an old man"  Even the "Mona Lisa" is sometimes mentioned as  the Golden Ratio is supposed to be found in the dimensions of a rectangle around Mona Lisa's face. In the absence of any clear (and documented) indication of where precisely such a rectangle should be drawn, this idea represents just another opportunity for number juggling.

The case of the two versions of "Madonna of the Rocks" (one in the Louvre in Paris, and the other in the National Gallery in London) is equally unconvincing.

The ratio of the height to width of the painting thought to have been executed earlier  is about 1.64 and of the later one 1.58, both reasonably close to (~ but also close to the simple ratio of 1.6.)

The dating and authenticity of the two "Madonna of the Rocks" also put an interesting twist on the claims about the presence of the Golden Ratio.

Experts who studied the two paintings concluded that, without a doubt, the Louvre version was done entirely by Leonardo's hand, while the execution of the National Gallery version might have been a collaborative effort and is still the source of some debate. The Louvre version is thought to be one of the first works that Leonardo produced in Milan, probably between 1483 and 1486. The National Gallery painting, on the other hand, usually is assumed to have been completed around 1506. The reason that these dates may be of some significance is that Leonardo met Pacioli for the first time in 1496, in the Court of Milan. The seventy-first chapter of the Divina (the end of the first portion of the book) was, in Pacioli's words: "Finished this day of December 14, at Milan in our still cloister the year 1497." The first version (and the one with no doubts about authenticity) of the "Madonna of the Rocks" was therefore completed about ten years before Leonardo had the opportunity to hear directly from the horse's mouth about the "divine proportion." The claim that Leonardo used the Golden Ratio in "Madonna of the Rocks" therefore amounts to believing that the artist adopted this proportion even before he started his collaboration with Pacioli. While this is not impossible, there is no evidence to support such an interpretation.

Either version of "Madonna of the Rocks" represents one of Leonardo's most accomplished masterpieces. Perhaps in no other painting did he apply better his poetic formula: "every opaque body is surrounded and clothed on its surface by shadows and light." The figures in the paintings literally open themselves to the emotional participation of the spectator. To claim that these paintings derive any part of their strength from the mere ratio of their dimensions trivializes Leonardo's genius unnecessarily. Let us not fool ourselves; the feeling of awe we experience when facing "Madonna of the Rocks" has very little to do with whether the dimensions of the paintings are in a Golden Ratio.

A similar uncertainty exists with respect to the unfinished "St. Jerome" (currently in the Vatican museum). Not only is the painting dated to 1483, long before Pacioli's move to Milan, but the claim made  that "a Golden Rectangle fits so neatly around St. Jerome" requires quite a bit of wishful thinking.

In fact, the sides of the rectangle miss the body (especially on the left side) and head entirely, while the arm extends well beyond the rectangle's side.

The last example for a possible use of the Golden Ratio by Leonardo is the drawing of  "a head of an old man." The profile and diagram of proportions were drawn in pen some time around 1490. Two studies of horsemen in red chalk, which are associated with Leonardo's "Battle of Anghiari," were added to the same page around 1503-1504.

While the overlying grid leaves very little doubt that Leonardo was indeed interested in various proportions in the face, it is very difficult to draw any definitive conclusions from this study. The rectangle in the middle left, for example, is approximately a Golden Rectangle, but the lines are drawn so roughly that we cannot be sure. Nevertheless, this drawing probably comes the closest to a demonstration that Leonardo used rectangles to determine dimensions in his paintings and that he might have even considered the application of the Golden Ratio to his art.

Leonardo's interest in proportions in the face may have another interesting manifestation. In an article that appeared in 1995 in the Scientific American, art historian and computer graphics artist Lillian Schwarz presented an interesting speculation. Schwarz claimed that in the absence of his model for the "Mona Lisa," Leonardo used his own facial features to complete the painting. Schwarz's suggestion was based on a computer-aided comparison between various dimensions in Mona Lisa's face and the respective dimensions in a red chalk drawing that is considered by many (but not all) to be Leonardo's only self-portrait.

However, as other art analysts have pointed out, the similarity in the proportions may simply reflect the fact that Leonardo used the same formulae of proportion (which may or may not have included the Golden Ratio) in the two portraits. In fact, Schwarz herself notes that even in his grotesques-a collection of bizarre faces with highly exaggerated chins, noses, mouths, and foreheads-Leonardo used the same proportions in the face as in the "head of an old man."

If there exist serious doubts regarding whether Leonardo himself, who was not only a personal friend of Pacioli but also the illustrator for the Divina, used the Golden Ratio in his paintings, does this mean that no other artist ever used it? Definitely not.

But the surge of Golden Ratio academic literature toward the end of the nineteenth century, initially did not start by projecting this idea in some of Da Vinci’s paintings, but rather by "The Parade of a Circus" (sometimes called "The Side Show") by the French pointillist Georges Seurat (1859-1891).

Seurat was interested in color vision and color combination, and he used the pointillist (multidotted) technique to approximate as best as he could the scintillating, vibratory quality of light. He was also concerned late in life with the problem of expressing specific emotions through pictorial means. In a letter he wrote in 1890, Seurat describes succinctly some of his views:

“Art is harmony. Harmony is the analogy of contradictions and of similars, in tone, shade, line, judged by the dominant of those and under the influence of a play of light in arrangements that are gay, light, sad. Contradictions are . . . , with respect to line, those that form a right angle.... Gay lines are lines above the horizontal; ... calm is the horizontal; sadness lines in the downward direction.”

Seurat used these ideas explicitly in "The Parade of a Circus"; currently in The Metropolitan Museum of Art, New York. Note  The entire composition is based on principles that Seurat adopted from art theorist David Sutter's book La philosophie des Beaux-Arts appliquee a la peinture (The philosophy of the fine arts applied to painting; 1870). Sutter wrote: "when the dominant is horizontal, a succession of vertical objects can be placed on it because this series will concur with the horizontal line."

Golden Ratio aficionados often present analyses of "The Parade" (as well as other paintings, such as "The Circus") to "prove" the use of  the Golden Ratio in Suerat’s painting.. Even in the beautiful book Mathematics, by Bergamini and the editors of Life Magazine. we find: "La Parade, painted in the characteristic multi­dotted style of the French impressionist Georges Seurat, contains numerous examples of Golden proportions." The book goes even further with a quote (attributed to "one art expert") that Seurat "attacked every canvas by the Golden Section." Unfortunately, these statements are unfounded.

This myth was propagated by the Romanian born prelate and author Matila Ghyka (1881-1965), who was also the "art expert" quoted by Bergamini. Ghyka published two influential books, Esthe­tique des proportions dans la nature et dans les arts (Aesthetics of proportions in nature and in the arts; 1927) and Le Nombre d'Or: Rites et rythmes pytagoriciens clans le developpement de la civilisation occidentale (The golden number, Pythagorean rites and rhythms in the development of Western civilization; 1931).

Both books are composed of semimystical interpretations of mathematics. Alongside correct descriptions of the mathematical properties of the Golden Ratio, the books contain a collection of inaccurate anecdotal materials on the occurrence of the Golden Ratio in the arts (e.g., the Parthenon, Egyptian temples, etc.).

The effect these speculations had on the esoteric-occult milieu of the 1930-1940’s, is most apparent in the speculations about  Poussin's painting  in “The Da Vinci Code” movie and book. (See  below)

The occult-esoteric author Waaler de (a Theosophist) claimed the builders of the Pyramids used the five pointed star as their meajurement as  the figure of Man. And when we realize that the Great Pyramid according to Cubic rivals the legendary city of Atlantis, we should not be too surprised to hear that.

The  theory appeared first in 1838, in Letter from Alexandria, on the Evidence of the Practical Application of the Quadrature of the Circle, in the Configuration of the Great Pyramids of Egypt, by H. Agnew. It is based on a (misinterpreted ) passage by Herodotus, by assuming that Herodotus meant that the number of square feet in each face is equal to the number of square feet in a square with a side equal to the pyramid's height.

Even with this "imaginative" interpretation, Agnew. Cubix, and today such authors as Graham Hancock and Robert Bauval  in a  book to be released in the UK Mai 2004, are still left with the small problem that the number mentioned (eight plethra) is way off the actual measurements. The suggested solution to this problem is even more appalling. With no justification whatsoever, H. Agnew claimed that the eight plethra must be multiplied by the area of the base of one of the small pyramids standing on the east side of the Great Pyramid. Typical for esoteric-occult thinking however Cubic added  that the Golden Ratio as a mystic symbol may have been deliberately hidden within the design of the Great Pyramid "as a message for those who understand."

The conclusion from all of this is that Herodotus' text can hardly be taken as documenting the presence of the Golden Ratio in the Great Pyramid. The totally unfounded interpretation of the text instigated by Agnew. Cubix (and subsequently Hancock/Bauval) really makes little sense and represents just another case of number juggling.

The fact that spherical drums were probably used to measure long distances during construction of the Pyramids would be sufficient to explain the above.

Other Egyptologists even claim that there does exist direct evidence suggesting that neither the Golden Ratio nor pi were used in the Great Pyramid's design (not even inadvertently).

This theory is based on the concept of the seked. The seked was simply a measure of the slope of the sides of a pyramid or, more precisely, the number of horizontal cubits needed to move for each vertical cubit. Clearly, this was an important practical concept for the builders, who needed to keep a constant shape with each subsequent block of stone.  The Rhind Papyrus deal with calculations of the seked have early on been described  in Richard J. Gillings's  book, Mathematics in the Time of the Pharaohs.

In any case the ‘Golden means’ was not discovered by an Egyptian but by the Greek Euclid, however not for building (it was never used in any Greek building including not the Parthenon as some occultists today still claim)  but  because he was interested in using this simple proportion for the construction of the pentagon and the pentagram.

The Golden means or Ratio is a product of humanly invented geometry. Humans had no idea, however, into what magical fairyland this product was going to lead them. If geometry had not been invented at all, then we might have never known about the Golden means.

Conclusion: Poussin's painting  in “The Da Vinci Code” movie and book.

From the evidence presented on this website it is clear that Pierre Plantard and his friend de Cherisey created their own  bloodline that started as a practical joke.

And the Priory of Sion was indeed formed in 1956 (as a joke among by now four, friends) but that does not mean that it had knowledge of Berenger Sauniere from that time on - no reference is made to Rennes-le-Chateou in any Plantard-related writings before 1964, and here it must be emphasised that Plantard first tried to exploit the ‘Gisors mystery’.

The X-Ray analysis of Poussin's painting 'Shepherds of Arcadia' proves that it was not based on any geometrical framework, as repeated by Professor Martin Kemp of Oxford University.

Originally Professor Cornford suggested that the Poussin painting 'Shepherds of Arcadia' was based upon pentagonal geometry - but that was only his opinion which he could not prove - and he could only make such a claim if he based his opinion on a geometrical structure that extended "outside" the boundaries of the canvas of the painting...

Had there been evidence in the X-Ray of the painting to support his claims relating to pentagonal geometry Professor Cornford would have used that material. But there is no such evidence to be found in the X-Ray of the painting.

This was all raised during the making of the 1996 television documentary 'The History of a Mystery' which debunked the theories of Andrews & Schellenberger and Henry Lincoln...Martin Kemp used Lincoln's own commissioned evidence of the X-Rays to dismiss the underlying geometry theory to Poussin's painting.

The reason why Poussin and Teniers were linked together in the fake 1960s PoS parchment code in fact is very simple - during the early 1960s when Plantard revived the Priory of Sion (having come out of prison) he claimed that it was founded by Godfroi de Bouillon during the Crusades.

Both Poussin and Teniers depicted scenes from Torquetto Tasso's Poem 'Jerusalem Liberated' - the central character of which was Godfroi de Bouillon.

When Plantard found the tomb at Les Pontils that was built in 1933 by Louis Bertram Lawrence to contain the dead bodies of his grandmother and mother, he incorporated Poussin's 'Shepherds of Arcadia' into the hoax, claiming that it was the "prototype" for Poussin's painting.

This mythmaking is extremely easy to put together and to fabricate- witness the British Israelite activities of the late 1900s...the Harp of Ireland became the Harp of David, etc. Martin Kemp has stated that all sorts of geometric 'coincidences' can be demonstrated from various paintings, but artists like Poussin may not have intended to depict what people are seeing in his paintings. Poussin (like virtually all other artists) left no indication that he based his works on geometric proportions.

Quoting Martin Kemp, Professor of Art History, Oxford University:

"If we look at a picture like 'The Arcadian Shepherds' by Poussin, it's easy to find what you're looking for in terms of geometry. I think the really important point from the standpoint of a historian like myself, is that we have hundreds and hundreds of drawings by Poussin - thousands and thousands including his contemporaries - now, not in one of those is there any evidence of a geometrical armature, of angles, of precise proportions being laid down - either at the deeper level or at the top level. It's not there."

Poussin's paintings have been analysed both with infra-red and X-Rays and these have shown no evidence of his paintings being based upon any geometric framework....

Professor Martin Kemp is not only Professor of Art History at Oxford University, but a leading authority on the use of mathematical and geometrical principles in painting.

Here is another quote from him:

"There are certain historical problems of which the Turin Shroud is one - in which there is fantastic fascination with the topic but a historical vacuum - a lack of solid evidence, and where there's a vacuum - nature abhors a vacuum - historical speculation abhors a vacuum - and it all floods in...

But what you end up with is almost nothing tangible or solid. You start from a hypothesis, and that is then deemed to be demonstrated more-or-less by stating the speculation. You then put another speculation on top of that, and you end-up with this great tower of hypotheses and speculations - and if you say "where are the rocks underneath this?" - they are not there.  
 

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April 9, 2004