SQUARING THE CIRCLE OF VITRUVIAN MAN
Presentation by Bro. David Walker Lodge of Happiness No. 7952
Brethren, by the kind invitation of the Worshipful Master I now present to you: ‘Squaring the Circle of Vitruvian Man,’ an illustration by Leonardo Da Vinci (circa 1487.) Like all of his notes, it was discovered some 300 after his death. Leonardo died in 1519.
To understand the task of Squaring the Circle in the context of this iconic image, it is important to first look back to its origins, which are firmly rooted in the Greek, Roman and Egyptian ancient civilisations.
1
MAN: THE ULTIMATE DIVINE CREATION Thutmose III - King of Egypt (1479-1425 BC)
Man has always been fascinated with ‘Man: the divine creation’ – created in God’s image and his proportions being a model for divinity in life and architecture. This image is of Thutmose III, King of Egypt (1479-1425 BC) At this time figures were generally draw using a grid, typically 18 squares high to the hair line or in this case 14, with the other 4 being accounted for by the seated position.
The figure of Thutmose III on this board was perhaps a preliminary drawing that was later to be transferred to a tomb or temple wall, while the other drawings were presumably practice hieroglyphs. The Right Forearm of Thutmose can be seen to be 4 grid squares long, the foot 3 squares, lower leg 6 etc. Each square corresponding to a ‘hand.’ 5 squares is an Egyptian Royal Cubit. From our ritual you will know of the term Cubit from the 2nd Degree Tracing Board. I will come back the Cubit shortly. But first I move forward another 1400 years to…
2
MARCUS VITRUVIUS POLLIO (70 - 15 BC) Vitruvius is the author of Da architectura, known today as The Ten Books on Architecture, written in Latin and Greek and dedicated to the emperor Augustus. Vitruvius is the Godfather of Modern Architecture. He was a Roman writer, architect, engineer and more besides. Book 1 teaches that a good Architect should be proficient in all of the liberal arts and sciences
1684 depiction of Vitruvius (right) presenting Da Architectura to Augustus
DE ARCHITECTURA, BOOK III (TEMPLES) In Chapter 1 Vitruvius states: “The design of Temples depends on symmetry the rules of which Architects should be most careful to observe. “Symmetry arises from proportion “Hence no building can be said to be well designed which wants symmetry and proportion “In truth they are as necessary to the beauty of a building as to that of a well formed human figure”
DE ARCHITECTURA, BOOK III (CONTINUED) Chapter 1 para 3 he states: “Just so the parts of Temples should correspond with each other, and with the whole. The navel is naturally placed in the centre of the human body. If in a man lying with his face upward, and his hands and feet extended, from his navel as the centre, a circle be described, it will touch his fingers and toes. It is not alone by a circle, that the human body is thus circumscribed, as may be seen by placing it within a square. For measuring from the feet to the crown of the head, and then across the arms fully extended, we find the latter measure equal to the former; so that lines at right angles to each other, enclosing the figure, will form a square.� Chapter 1 was solved by Leonardo Da Vinci 1500 years later
DEPICTIONS OF VITRUVIAN MAN (BY CESARE CESARIANO) IN THE 1521 EDITION OF DA ARCHITECTURA
FOR 1500 YEARS THE GREAT AND GOOD FAILED TO ILLUSTRATE VITRUVIAN ANATOMY
Every attempt to illustrate the profound words of Vitruvius had failed miserably. Cesare probably came closest in his illustrations in 1521 but even he didn’t match Leonardo. Individual images were produced of a man circumscribed by either a Square or a Circle, but not both concurrently. When both were used simultaneously, the results distorted the proportions that Vitruvius so explicitly specified, creating alien or freakish images of a man, and certainly not an archetypal divine model. Most failed this aesthetic test. Vitruvius may have intended that a man should be separately circumscribed by a circle and square. However, a conjoined Square & Circle has long been sought as the ideal blend of body, mind and soul. Most attempts to illustrate Vitruvian Man failed simply because they used the Naval as the centre for both the square and circle – as it turns out because of a simple miss reading of Vitruvius, as he didn’t describe the centre of the square in this manner. Cue Vitruvian Man…
6
LEONARDO DA VINCI’S VITRUVIAN MAN "Man is the model of the world.“ Leonardo Da Vinci Leonardo famously used mirrored text in his note books. Some say to hide is discoveries, others say because he was left handed and wanted to avoid smudges. In this iconic image the mirrored writing contains no secrets, as it just quotes Vitruvius directly. If there were no secrets then perhaps the whole page is mirrored? The Upper Text first refers to proportions: “Vitruvius, the architect, says in his work on architecture that the measurements of the human body are distributed by Nature as follows: •4 fingers make 1 palm •4 palms make 1 foot •6 palms make 1 cubit •4 cubits make a man's height •4 cubits make one pace •24 palms make a man; and these measures he used in his buildings.”
Then the Circle: “If you open your legs so much as to decrease your height 1/14 and spread and raise your arms till your middle fingers touch the level of the top of your head you must know that the centre of the outspread limbs will be in the navel and the space between the legs will be an equilateral triangle.”
1 Palm is ~ 3” or a course of modern brick work {demonstrate units of measurement} 1 Cubit equals 24 Fingers A Pace is a Roman Pace which is two strides, measured from the back of heal to the point where the same heal strikes the floor. An Inch does not fit within this anatomical measuring system, being a more modern English invention. An Inch equates to the width of the thumb or literally 1/12th of a foot. Our 24” gauge is the modern equivalent of a cubit, but is between 4 & 6” longer than a Standard Cubit, as it is 24 Thumbs long rather than 24 Fingers. A Royal Cubit is a Short Cubit + 1 palm or 4 fingers. The Royal Cubit was used in temples as a more holy or godlike proportion. Each civilisation had a different Cubit based on their own standardised anatomy. Basically, a Short Cubit is ~ 17.5” , a Royal Cubit is ~ 20.5”, with each in Egyptian, Greek, Roman, Hebrew and Babylonian forms and the Standardised Cubit of ~ 18”.
These ancient units are based on the proportions of the body and not strict standardised lengths. Even today, even though modern man is bigger than our ancient brethren, the proportions of the body remain the same. The Architect or Master Mason would have been in charge of standardising the units of measurement and tools used in construction. These units would have been universally understood, right from the Master, through the Craftsmen, to the Apprentices. {demonstrate signals} Can I ask W Bros, Scales, Willets and another volunteer to assist me with a practical demonstration? {demonstrate proportions, circle}
7
LEONARDO DA VINCI (LOWER TEXT) Lower Text then refers to the Square: •The length of a man's outspread arms is equal to his height. •From the roots of the hair to the bottom of the chin is the 10th of a man's height; •from the bottom of the chin to the top of his head is one 8th of his height; •from the top of the breast to the top of his head will be one 6th of a man. •From the top of the breast to the roots of the hair will be the 7th part of the whole man. •From the nipples to the top of the head will be the 4th part of a man. •The greatest width of the shoulders contains in itself the 4th part of the man. •From the elbow to the tip of the hand will be the 4th part of a man; •and from the elbow to the angle of the armpit will be the 8th part of the man. •The whole hand will be the 10th part of the man; •the beginning of the genitals marks the middle of the man. •The foot is the 7th part of the man. •From the sole of the foot to below the knee will be the 4th part of the man. •From below the knee to the beginning of the genitals will be the 4th part of the man. •The distance from the bottom of the chin to the nose and from the roots of the hair to the eyebrows is, in each case the same, and like the ear, a 3rd of the face.
HOW LEONARDO CRACKED IT Squaring the circle is used in general language as a metaphor for attempting the impossible. For 1500 years the task of illustrating Man‟s proportions as described by Vitruvius had proved impossible, yet Leonardo Da Vinci, one of the greatest men of the renaissance, cracked it in 1487. Not only did he crack the problem, he did it so well that his iconic image has become possibly the most famous and most copied image in human history. The great pity is that his notes weren‟t discovered until some 300 years after his death. To understand how successful Leonardo was, where others had so spectacularly failed, it needs an understanding of the nature of the man. Remember that Vitruvius said that: “a good Architect should be proficient in all of the liberal arts and sciences.” Nobody can question that Leonardo was the worlds greatest student & Master of the Liberal Arts & Sciences. He was an exceptional visionary, writer, teacher, scholar, artist, mathematician, sculptor, inventor, engineer, geophysicist anthropologist and anatomist. In the ancient world the role of an Architect was different to what we understand it to be today. The word Architect is derived from the Greek word arkhitekton This itself is derived from arkhi meaning "chief, first, principle, or master“ and tekton meaning “craftsman, builder, carpenter or mason – a generic term applied to any craftsman.“ In this context it wouldn‟t be unreasonable to describe both Vitruvius & Leonardo as Master Masons. Clearly, Leonardo was the only person who had a broad enough understanding solve the problem.
SQUARE & CIRCLE DEFINED BY THE GOLDEN RATIO
SQUARE & CIRCLE DEFINED BY A ROTATED SQUARE
HE DIDN’T QUITE USE THE GOLDEN RATIO
Contrary to several propositions I have read, Leonardo did not use the Golden Ratio or Golden Section (namely, 1.618 or its inverse of 0.618) to construct his illustration. There are several puncture marks in the original that show he was experimenting with the position of his radial point. I have rectified the image to remove camera tilt, perspective & lens distortions. The above measurements have been measured in the Computer Aided Design Software I use at work (AutoCAD) The diagram on the left shows how his illustration would look using the Golden Ratio to define the radius of the circle. To find the golden ratio from just a square with a width of 1, using a set of compasses, you merely have to draw an arc, centred on the middle of the top of the square, who’s radius is fixed by a bottom corner. Draw the arc from the corner back up to the projected line of the top of the square. The distance of the projected line is 0.618 which is a Golden Ratio of the square. The circle is then drawn using this distance as the radius. The circle is bigger than in Leonardo’s illustration, so the fingers can not touch the circle, failing Vitruvius' instructions.
By contrast the diagram on the right is how Leonardo established his radius. Turn a copy of the square by 45 degrees about the centre of the square., to create an octagram . Leonardo’s circle is from the top of the rotated square to the base of the original square and has a radius of 0.604 – nearly the golden ratio, but out by a factor of 2 fingers. Some reports suggest that the Ancient Greeks used 1.6 or 0.6 as the Golden Ratio. Like many of his works, this work may be unfinished, particularly when you consider that the position of the left most leg isn’t is the correct position to create an equilateral triangle. However, it is fair to say the illustration does approximate the golden ratio and that approximation does appear several time in the proportions of the body.
10
SYMBOLISM OF VITRUVIAN MAN Squaring the Circle, in the geometrical sense, has been challenging the minds of mathematicians and geometers since Greek Times. The concept has probably been challenging man for long before that. The phrase „Squaring the Circle‟ can itself be traced back to 414 BC. In 1882 Squaring the Circle was proven and is now accepted to be impossible, as it requires the construction of the Square Root of Pi with just a straight edge and compasses. Whilst Leonardo wasn‟t attempting to Square the Circle in the geometrical sense he was metaphorically. Squaring the Circle is used as a metaphor for achieving the impossible or finding a simple solution to a complex problem. He used all of his knowledge of geometry, mathematics, art and anatomy to Crack Vitruvian Proportion. Doing so in a clever, yet simple way. Vitruvius taught the world the importance of harmonic proportions, echoing man‟s proportions in the built world. Leonardo Da Vinci showed the world the true import of Vitruvius and is so doing, gave us the ultimate marriage of Architecture, Anatomy & Geometry and the perfect answer to the riddle of Squaring the Circle, in a powerful and beautiful way. Both Leonardo and Vitruvius were Masters of the Liberal Arts & Sciences, both arkhitekton and Master Masons in the operative sense. As a Speculative Master Mason what follows is entirely my speculative take on the symbolism of Vitruvian Man. Previously I said that the whole illustration may have been mirrored and not just the text. Now I suggest that it is mirrored as it holds a secret. As Freemasons we are taught through our ritual that the VSL governs the soul, the circle the mind and the square the body; that the square regulates our actions and the compasses keep us in due bounds. We are encouraged to study the Liberal Arts and Sciences as may lie within the compass of our attainment, so that we may estimate the wonderful works of the Almighty. Through square conduct level steps and upright intentions we hope to ascend to the immortal mansion. Having recently undertaken the 3rd degree working tools it brought me back to something that has fascinated me for a long time. That of geometry in my profession as a surveyor, Vitruvian Man, and the great works of Leonard Da Vinci. However, my journey into Freemasonry has given me further pause for thought and led me to re-evaluate the those things. Whilst learning the 3rd degree working tools the image that was in my mind was always of Vitruvian Man.
Point Within a circle
Whilst there are no parallel lines or VSL, I still see them in my minds eye. I see a man centred within the circle using the compasses to measure his own conscience, to define his due bounds and at a point from which he cannot err. A man Guided by the VSL and following in the steps of Moses and King Solomon
4 Cubits Makes a Man
The square is 4 cubits high and wide, defining the limits of a Man and the width of the Pillars at the entrance to King Solomon's Temple; I see a foursquare man, solid, strong, forthright & honest, standing perfectly erect, his feet formed in a square, showing that straight and undeviating line of conduct laid down in the VSL
7 is the number
The spread eagled Man is in the shape of an implied Heptagon with 7 sides. Which reminds me of: 7 to make a Lodge Perfect - opened on the centre, 7 years to build King Solomon's Temple and I see a man at one with the 7 liberal arts & sciences, estimating the works of the almighty, within the compass of his attainment.
Drawn with nothing more than the 24� gauge, compasses and pencil.
ARKHITEKTON MAN A Master Mason Standing in a temple: Prepared with the 24” Gauge, Common Gavel and Chisel Proofed with the Square, Plumb Rule and Level Planned with the Skirret, Pencil & Compasses
A Master of the Liberal Arts & Sciences Conscientious, Moral, Ethical, Centred & Level headed, who having squared his circle is a perfect blend of mind, body & soul. We dance round in a ring and suppose… But the secret sits in the middle and knows. Short poem by Robert Frost
Thank you Brethren for listening to my speculative take on Vitruvian Man, it’s operative origins and what is says to me as a Surveyor and Freemason. I will leave you with this short poem by Robert Frost…. We dance round in a ring and suppose… But the secret sits in the middle and knows. Hopefully I have Squared the Circle for you.
13
REFERENCES/FURTHER READING: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)
The British Library - Thutmose III BBC science website – Leonardo Da Vinci BBC video - The Beauty of Diagrams: Vitruvian Man University of St Andrews – Squaring the Circle by J J O'Connor and E F Robertson Virtuvius - De Architectura (online versions by Bill Thayer &/or Morris Hicky Morgan) Museo Galileo / Institute & Museum of the History of Science / 28th Exhibition of the European Council entitled Universal Leonardo – The Mind of Leonardo World-mysteries.com - Geometrical Construction of Vitruvian Man www.chateau-de-mezerville.org – Historical Geometrical Curiosities leonardodavinci.stanford.edu – The Worlds of Leonardo Da Vinci course material & projects Wikipedia – various related to all sections D.M. Murdock - Jesus Christ Mason of God Isaac Newton – A Dissertation upon the Sacred Cubit of the Jews and The Cubits of the several nations www.masterbuilderfellowship.com/page5.html - The Ancient Masterbuilder By Stephen Parcell - Four Historical Definitions of Architecture John Legon – The Cubit & the Egyptian Canon of Art Clive Ross – The Royal Cubit and the Cubit Measure