Vitruvius was an ancient Roman architect who wrote a series of ten books on architecture - one of the few collections of books of its type that survived into the Renaissance. In the third volume, which is on the proportions of temples, he states that these buildings should be based on the proportions of man, because the human body is the model of perfection. He justifies this by stating that the human body with arms and legs extended fits into the perfect geometric forms, the circle, and the square.

This fragment of the philosophy of Pythagoras seized the imagination of the Renaissance. Many artists tried to illustrate this divine relationship, but with varying success. An illustration of Vitruvian man by Cesariano in his Cosmo Vitruvius of 1521 reeks of failure. Cesariano drew a perfect circle and square tangent to each other at the four points of the square; then he forced a figure of a man into the design so that his hands and feet touch the points. The result was one of the most disproportioned figures of the Renaissance, with arms too long, legs too short, and hands and feet too big. A system of relationships alone did not make beauty happen.

It took the genius of Leonardo da Vinci to solve the problem. Leonardo started by drawing a perfectly proportioned man and then found the circle and square in the figure. The circle and square are only tangent at one place, the base. The thing that he added was beauty.

Some of you will be asking yourself, why was it so important to the Renaissance artists and philosophers that a human body could fit into a circle and a square? Some of you probably know the answer, but I will go into it for the benefit of those who do not.

The ideas that Vitruvius was expressing can be traced back to Pythagoras. Pythagoras lived in a Greek colony in Southern Italy in the 6th century, BCE, the same time that Buddha lived. Like Buddha, Pythagoras taught his male and female disciples that life is an endless wheel of reincarnations until we purify ourselves and return to our divine source.

Purification included a vegetarian diet, moral behavior, and contemplation of the numerical abstractions that underlie reality. Pythagoras was the first person to call himself a philosopher, which means to love Sophia (wisdom). We have no writing that can be attributed to Pythagoras; yet, he is one of a handful of people that were instrumental in creating Western culture. It is theorized that if he did write anything it would have been poetry and he would have signed it Orpheus.

Orpheus, the mythical, semi-divine musician, was the founder of the first mystery cult, a religion based on a secret redemptive ritual. This religion is believed to be a major source for the Pythagorean teachings. Many of its followers were poets and musicians who believed that their inspiration came directly from Orpheus; hence, they would sign his name to their work.

In the Orphic creation myth, the beautiful god, Dionysus, is born of the incestuous union of Zeus and Persephone. Zeus' wife, Hera, is jealous and wishes to destroy the child. To accomplish this she has her allies, the Titans, dismember and devour him. Of course, Zeus is heart broken and in a fit of anger, he burns the Titans to ash with a volley of lightning bolts. Only Dionysus's heart remained, and from this, Zeus creates a new Dionysus. However, from the ash of the Titans mixed with the devoured Dionysus, the human race was born. Therefore, the human race is part divine and beautiful like Dionysus and part vicious and material like the Titans. The purpose of the Orphic mystery was to redeem the Dionystic soul and make it the dominant influence in the lives of the devotees.

The Orphics, like Pythagoras, saw a connection between music and numerical order. This type of reasoning lead to sacred geometry. Pythagoras taught that numbers had qualities as well as quantity and that geometric figures were powerful magical symbols. The circle, being connected to the dome of the sky and the cosmos with its spherical stars and planets continually circling the earth was a symbol of Dionysus, the soul.

The square, on the other hand, is the natural way that humans relate to the physical world. This is why there are four directions, four seasons, and four elements. It is why my house has four sides and I am sitting on a four-legged chair while I write this on my square keyboard and read it on my square screen. The square was a symbol of the Titanic human aspect.

The first step to the liberation of the soul is to recognize that we are made of both aspects. In Pythagorean thinking, if a human can be shown to fit into both symbols this would be a geometric proof of our dual nature. Many of these teachings were incorporated into alchemy, and other ancient disciplines. In this way, the teachings - although at times fragmented - were passed on to the Renaissance. In Venice around the year 1500, Leonardo once again demonstrated geometrically that the human soul is divine.

The common understanding of Pythagorean number mysticism seems to be that the Pythagoreans thought of the whole numbers as symbols that were equated to specific qualities as well as quantities - one was unity, two polarity, three the beginning of form, etc. This is true; however, the most powerful way that numbers were seen to impact on reality was as ratios. Ratios are relationships between numbers. For example, we can see that 4 relates to 8 the same way that 1 relates to 2 - they are the same ratio. The Pythagoreans found that music could be expressed as ratios, that the ratio 1:2 described the whole note, 2:3 the perfect fifth, and 3:4 the perfect forth, These realities underlie all musical harmony. Every culture has to find these same notes and create their musical scale around them in order to have music. Beauty has this objective aspect as well as a subjective aspect. Pythagoras believed that the universe was ordered in this same way. Because there were seven planets that he knew of, he created seven notes to correspond to them. He called the universe a cosmos, which meant that it was a beautiful, musical harmony - music of the spheres.

Classical artists used the same principal to develop the perfect figure, and ratio is what Leonardo made use of also. By this, we mean that we have to find a way of measuring the figure by making use of relationships within the figure - not by measuring it with an external scale such as inches or any other external system. The unit of choice was the head. Leonardo's figure has a one 1:8 ratio with its own head. In other words, it is 8 heads tall.

You will notice that Vitruvian Man has dividing lines drawn on his body. There is a line at the chin that indicates the limit of the 1 head unit, a line at the nipples that marks the length of 2 heads, a line at the groin that marks 4 heads, a line below the knees that marks 6 heads, and the base line the marks 8 heads. The base line forms the bottom of the square and the top rests on his head. Therefore, the square is 8 heads tall. Notice that the man's lower set of extended arms touch the square on both sides. Because the width of a square is equal to its height, the length of the arms has to be 8 heads as well. In other words, our extended arms are the same length as our body from head to toe. Try measuring your friends; you will find that this is true with only a slight variation. The vertical lines on the shoulders of Leonardo's figure measure the 2 head width of the torso at the shoulders; the line at the joint of each arm measures an additional 1 head in each direction; and then we jump another 2 heads on each side to the fingers and the sides of the square. However, the dividing line between the hand and the forearm stems from a different unit of measure. This is called the "golden proportion." Proportion is a relationship between two ratios. The Pythagoreans noticed that there are three different levels of complexity of proportion.

The most complex described a relationship of four items: A is to B as C is to D; 4:8 = 1:2. This was called discontinuous proportion. Next was continuous proportion, which involved three items: A is to B as B is to C; 1:3 = 3:9. The most sacred proportion involved two items, thereby drawing us back to primal duality. This was called the Golden Portion: A is to B as B is to A+B. Expressed mathematically as 1to the irrational number 1.61803commonly referred to by the Greek letter Phi. In Vitruvian Man, the length of his hand relates to his forearm in the same way that his forearm relates to the length of the forearm and hand combined- the golden portion.

These measures are then related to other parts of the figure. Notice how Leonardo conveniently shows us the man's left foot in profile. He even places the heel in front of the big toe of the right foot so that we can see the full length. The length of the foot is the same as the length of the forearm - demonstrating a golden relationship between the hand and the foot. The length of the hand can be related to the measurement of the face - from the chin to the hairline. The face in turn is divided into thirds, which coincide with the eyebrows and the tip of the nose - echoed in the length of the ears. The line at the eyebrows is the golden division of the entire head.

The golden relationship is the key to the proportions of the entire figure. For example, if we examine the length of the arm from the top of the shoulder to the tip of the fingers, we find that the elbow is the golden division. If we examine the length from the elbow to the fingers, we find the base of the hand at the wrist is the golden division. If we measure the hand, the fingers begin at the golden division, and fingers themselves are composed of segments with a golden relationship to each other. We can find the same pattern of proportion in the legs, feet, and torso. However, the most dramatic illustration of the golden portion is the division of the body at the navel. The relation ship of the distance from the top of the head to the navel is to the distance from the navel to the feet the same as the navel to the feet is to the entire height of the figure.

If we divide the square in half, we will find that the middle falls right at the line drawn through the groin. At the level of sexuality, we are centered in the physical. To find the center of the circle, the spiritual, we have to move up to the navel - the golden division. If we extend the horizontal line at the groin until it divides the square into two equal rectangles and then we draw diagonal lines connecting the corners of these rectangles. We will find that the diagonals intersect the circle as it extends through the center of each hand and foot. The pattern that is formed, with four equidistant points radiating out of a central fifth point (the navel where we were once connected to our life giving mother) is called a quincunx. This is a powerful archetypal symbol for sacredness that can be found in all cultures, and this is the same symbolic structure that is found on the World card in the Tarot.

The World card in the Tarot illustrates the same squaring of the circle. It depicts a sacred figure in the center; the archetypal Great Mother identified with the Navel of the World. To the alchemists, Hermeticists, and Neoplatonists, she is called the Anima Mundi, the Soul of the World. She forms a quincunx in relationship to the four figures in the four corners of the card - the four Evangelists, who through their relationship to the four fixed signs of the Zodiac relate to the four elements, the four directions, and the four seasons. The entire Tarot deck shares in this structure. The four minor suits relate to the square - the four elements, seasons, and directions. The trumps that move us along a spiritual journey of triumphs, that start with the four fold mundane world, take us through the hero's journey of death and rebirth, and back to the world - only now with the spirit exposed - the naked truth - are the circle. The message of the Tarot is the squaring of the circle.

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