The Vitruvian Man is a drawing by Leonardo da Vinci. The study sheet shows the ideal proportions of a man as described by the ancient architect Vitruvius. The drawing is in the museum Galleria dell' Accademia in Venice and is rarely exhibited publicly for conservation reasons.
Leonardo da Vinci's paintings cannot be understood without examining the geometry of the picture
Leonardo da Vinci
Pen and ink on paper
34,4 × 24,5 cm
Galleria dell’ Accademia, Venice
A sheet of paper. In the sheet center a drawing. An older naked man with curly hair is shown in frontal view. Arms and legs are shown in two views each. They are arranged so that the arms stretched out sideways and the man's body height span a square.
A second view superimposes the first. It shows the man's raised arms and spread legs. The navel forms the center of a circle that completely encloses the man. Here the man, circle and square have three points in common on the middle finger of the raised hands, as well as on the right toe. On the man himself there are more strokes that divide his body into sections.
Below the drawing a horizontal line is shown. It is as wide as the square spanned by the man and symmetrically divided by transverse strokes.
Above and below the drawing are handwritten notes in Italian. The texts are written in mirror writing. In the upper part, the author has led the writing around the circle. In the lower right corner the drawing is signed "Lionardo da Vinci".
The sheet is in good overall condition, but has stains and discoloration here and there.
Leonardo's inscription - the original text
For a better overview, Leonardo's text is divided into paragraphs here.
Above the drawing
"Vitruvius, the architect, in his work on architecture, says that the measurements of the human body are distributed by nature as follows:
4 fingers make a palm,
4 palms make a foot,
6 palms make a cubit;
4 cubits make the height of a man. And
4 cubits make a step, and
24 palms make a man;
and these measurements he used in his buildings.
[Here the quotation ends and Leonardo begins with his own observations].
If you open your legs so wide that they reduce your height by 1/14, and spread and raise your arms until your middle fingers touch the height of your head, you must know that the center of the spread limbs is at the navel, and the space between the legs is an equilateral triangle."
Below the drawing
"The length of a man's outstretched arms is equal to his height.
From the hairline to the underside of the chin is one-tenth of a man's height;
from the bottom of the chin to the top of the head is one-eighth of a man's height;
from the top of the chest to the top of the head is one-sixth of a man's height.
From the crown of the chest to the roots of the hair is the seventh part of the whole man.
From the nipples to the crown of the head is the fourth part of a man.
The greatest width of the shoulders contains within itself the fourth part of a man.
From the elbow to the tip of the hand is the fifth part of a man,
and from the elbow to the armpit is the eighth part of man.
The whole hand is the tenth part of man;
the beginning of the genitals marks the middle of the man.
The foot is the seventh part of man.
From the sole of the foot to below the knee is the fourth part of man.
From below the knee to the beginning of the genitals is the fourth part of the man.
The distance from the bottom of the chin to the nose and from the hairline to the eyebrows are each the same, and like the ear, one-third of the face."
The ancient architect Vitruvius
The idea for this study of proportions did not come from Leonardo himself, but from the ancient architect and engineer Marcus Vitruvius Pollio, known as Vitruvius (c. 80-70 BC to c. 15 AD).
Vitruvius' main work "Ten Books on Architecture
Hardly anything is known about Vitruvius' life. Only his work "Ten Books on Architecture" (De architectura libri decem) has been preserved through numerous copies. Today it is considered the only surviving book on architecture and civil engineering from the time of Greco-Roman antiquity. In addition, according to Vitruvius, it was the only summary of the art of building at that time. It also contains a few biographical details of the author.
The work mainly deals with urban planning, materials science and water supply. The term architect was broader for Vitruvius. Thus, there are also two chapters on astronomy, time measurement and mechanical engineering.
The shapely body
In the third chapter, that of temple construction, Vitruvius introduces the doctrine of the well-formed body (homo bene figuratus). He recognized in human proportions cosmic rules that should be applied to all human works, such as architecture.
He stated that the center of the body is located in the navel, as well as that the height of man is equal to his width with arms stretched out sideways.
The basic unit of Vitruvius' theory of proportion was the width of a finger. Based on this, he laid down the rules to which Leonardo refers with his drawing:
1 finger width = 1/96 of the total size
1 hand span = 1/24 of the total size
1 hand span = 1/8 of the total size
1 forearm = 1/4 of the total size
1 foot = 1/6 of the total size
The data are more extensive in Vitruvius' book and are only excerpted here.
Rediscovery during the Renaissance
Vitruvius' writings were increasingly read and received during the Renaissance. Leonardo also owned a copy and compared it with the results of his own studies of proportion.
Surviving editions of Vitruvius' work are not illustrated, which led to various visual interpretations of the Vitruvian model. In particular, it was unclear where the center of the square encompassing the man should be. When compared with other contemporary interpretations of Vitruvius' rules, the quality of Leonardo's contribution becomes clearer.
Leonardo deviates in some details from Vitruvius' specifications. Basically he keeps to the finger width as a basic unit and also uses 96 units in width and height. But he does not set the width of the hands to 1/8, but shortens it to a more aesthetic 10/96. Analogously the length of the foot: Vitruvius gives 1/6 of the height, Leonardo shortens it to 1/7.
In this respect, Leonardo's drawing is not a mere illustration of Vitruvius' specifications, but a critical reception that does not shy away from incorporating its own findings from anatomical and mathematical studies.
Date of origin and owner
Nothing is known about the time of origin of the leaf. Likewise, there are no contemporary sources that prove its existence. The authorship of Leonardo is nevertheless considered by the experts as undisputed. The focus is on a book by the mathematician Luca Pacioli. Leonardo's Vitruvian Man is said to have been created as an illustration for this book.
Luca Pacioli's Vitruvian Studies
Like Leonardo, the mathematician Luca Pacioli was concerned with human proportions and the teachings of Vitruvius. Luca Pacioli was a student of the exceptional painter Piero della Francesca, and later an important mathematician. He wrote numerous treatises on geometry. Still significant today is Pacioli's book on double-entry bookkeeping, which is the first complete summary of this method. He also wrote a summary of the algebra known at that time.
Leonardo's collaboration with Pacioli
Around 1498 Luca Pacioli was invited by the Duke of Milan Ludovico Sforza. Leonardo was staying at the duke's court at the same time, and this is how he met Luca Pacioli. They began a collaboration that lasted several years and extended beyond the two years they spent together in Milan.
Among other things, Leonardo made 60 drawings of geometric solids for Pacioli's planned book "Divina Proportione" ('Divine Harmony'). In addition, he probably illustrated a now lost book on chess.
Paciolis Buch "Divina Proportione"
Pacioli completed his later very influential book around 1498. Initially, only three manuscripts were handmade to serve as particularly beautiful specimen copies for noble backers; one went to the Duke of Milan.
In the book, Pacioli deals with the laws of geometric solids, the golden section, and applications in architecture. In addition, he largely adopts essays by his teacher Piero della Francesca and explains numerous passages from Vitruvius' main work "Ten Books on Architecture". Among other things, the passage that is also quoted by Leonardo on the sheet with the Vitruvian Man.
Plato's idea of the microcosm and macrocosm
This shows that Leonardo's study sheet is not to be considered on its own, but the sheet is part of a broad examination by the scholars of the time of the ideas of the ancient masterminds, especially those of the scholar Plato.
Plato, Vitruvius, Pacioli and also Leonardo believed that all natural forms were based on simple geometric proportions. No matter if it was about the proportions of the planets or the length of their orbits (macrocosm). Or whether it was about the manifold forms of plant blossoms, insect bodies up to the smallest parts, the atoms (microcosm). And all these proportions were according to the ancient teachings in the greatest possible harmony. Integer proportions like halves, quarters and eighths were just as much a part of the explanation of the laws of nature as irrational number ratios like the circle number Pi, the golden section or the square root of 2 (known today as Din A4 format).
The human being, as a part of nature, was, according to the conception of that time, also created in harmonic proportions. Even if humans look different by small deviations, they are nevertheless on the average of the same shape. Just as a horse can be clearly recognized by its proportions, the appearance of a cat or a sparrow is also determined by the proportions of the limbs of their bodies. According to Vitruvius, these "Divine Proportions" (the German title of Luca Pacioli's book) were then to be applied to all architectural designs.
Leonardo was completely familiar with the ancient idea of a harmoniously proportioned universe and always showed this in his paintings. That his Vitruvian Man in a revised form (i.e. without handwritten explanations) was to become part of Luca Pacioli's book is conceivable, but cannot be proven. Leonardo's drawing would certainly have complemented Pacioli's explanations of Vitruvius' theory of proportion well.
Other proportion studies
The Vitruvian Man is not Leonardo da Vinci's only study of proportions. As his notebooks show, he often dealt with the subject.
In the possession of Francesco Melzi
It is believed that after Leonardo's death, the Vitruvian Man became the property of his student Francesco Melzi (1491-1570). Melzi accompanied Leonardo until the latter's death and then became the executor of his will as designated by him. He began to sort the numerous writings, called codices, thematically. One result of this work is Leonardo's "Book of Painting." After Melzi's death, the codices were scattered all over Europe. In the process, they suffered great damage. Some were cut up into individual pages, and in some cases even the individual leaves were cut up even further. Many sketch pages were also lost. Experts estimate the original volume of the codices at 20,000 pages, but today only about 6,000 of them remain.
In the possession of Giuseppe Bossi
What is known today about the whereabouts of the Vitruvian Man comes from the diary of the Italian artist Giuseppe Bossi (1777-1815). Bossi wrote in his diary that he had acquired the sheet in 1807, among many other Leonardo drawings, from the de Pagave family. The family, in turn, had received the drawings from Countess Anna Luisa Monti, heiress of Cardinal Cesare Monti (1594-1650). Beyond Bossi's statements, the origin of the sheet cannot be proven with sources.
Bossi's importance for the history of art
Bossi was himself a famous painter who carried out various commissions for the Napoleonic occupiers of Italy. Among other things, he was the artistic director for the production of a copy of Leonardo's famous Last Supper.
Bossi was also an art collector. He was especially interested in the works of the Renaissance and in particular in Leonardo, Raphael and Michelangelo. He was also involved in publishing the famous artist's vitae of Giorgio Vasari. The biography of Leonardo contained therein is today one of the oldest sources for Leonardo research.
Bossi is not without controversy, as he is said to be responsible for various art forgeries. Thus numerous drawings, which are considered today as Leonardo or Raffael drawings, are to come actually from him.
Bossi died of tuberculosis in 1815, the year of Napoleon's final abdication, at the age of only 37.
In the possession of the Galleria dell' Accademia
After Bossi's death, the print was initially auctioned off by his heirs. Seven years later, in 1822, it was acquired by the Viceroy of the Kingdom of Lombardo-Venetia, which was under Austrian rule. Since then it has been in the Galleria dell' Accademia in Venice.
The print is rarely shown in public for conservation reasons.
Geometry - angles, shape and proportion
The Vitruvian man is an illustration of human proportions, that is, it shows the geometrical relationships of its external limbs.
The line under the drawing functions as a scale for the proportions of the Vitruvian Man. The proportions of the drawing above are derived from its divisions.
The scale is divided into 96 units of equal size, corresponding to the width of a finger. It is symmetrically mirrored in the middle, so that the lines on the left and right correspond to the usual length units on the forearm at the time of the Renaissance.
- the smallest unit is the width of a finger (far left)
- four finger widths are the width of a palm (without thumb)
- six palms make one cubit
A cubit (Ital. 'braccio') was a medieval measure of length. It meant the length from the elbow to the fingertips. In the Middle Ages, measurements were made with standardized body units. In English-speaking countries, distances are still measured in feet.
I The perimeter
Leonardo has the man raise his arms so that his middle fingers form a horizontal line with the height of the man's head (upper white horizontal).
- "If you open your legs so that they reduce your height by 1/14, and spread and raise your arms until your middle fingers touch the height of your head, you must know that the center of the spread limbs is in the navel [...]"
The center of the circumcircle is in the navel (blue dot).
The feet are raised 1/14 of the height of the body (lower white horizontal), that is, half of 1/7.
- "From the crown of the chest to the roots of the hair is the seventh part of the whole man"
- "The foot is the seventh part of the man".
The lines of the seventh are highlighted in red.
The equilateral triangle
- "When you open your legs so wide, [...] you must know that [...] the space between the legs is an equilateral triangle"
The man's spread legs form approximately an equilateral triangle.
Leonardo deviates twice from the text in the drawing. The feet are not at the same height because the right leg has been pulled inward by about 5°. Thus the height of this foot is not reduced by 1/14 (lower white line).
Since the right leg was pulled inward, the space between the legs is not symmetrical. So there is no equilateral triangle. For that, the right leg would have to be further up. The comparison becomes clear with mouseover. Why Leonardo deviated from the text in these details is unknown.
The sevenths cannot be transferred to the scale, because 96 finger widths are not divisible by 7 without remainder. This applies accordingly to 14.
II The quarters
The square around the man is the starting point of a continuous subdivision by bisections
- "The length of the outstretched arms of a man is equal to his height".
The sentence is the most famous of the sheet. It serves as the heading of the text in the lower part.
The square around the man is divided horizontally into two halves
- "the beginning of the genitals marks the middle of the man"
The half of the surrounding square is again divided in half, creating quarters
- "From the nipples to the crown of the head is the fourth part of a man"
- "The greatest width of the shoulders contains within itself the fourth part of a man"
- "From the sole of the foot to below the knee is the fourth part of a man"
"From below the knee to the beginning of the genitals is the fourth part of a man"
Leonardo made the quarterings visible by strokes on the Vitruvian man (blue lines). They thus emphasize the observations made by Leonardo in the text.
The quarters can be bisected into eighths, which occur around the head (red lines).
- "from the elbow to the armpit is the eighth part of the human being"
- "from the bottom of the chin to the top of the head is one-eighth of the height of the body".
Reference to the scale
The constant division of the side length of the square by 2 can be directly applied to the scale below the drawing. The blue line shows the half, to the left and right of it the quarters and if these were divided, it would be the eighths or the width of three palms. This proportion is called 1:2 or 2:1.
At this point it is noticeable that the drawing contains two misrepresentations.
- the distance from elbow to armpit on both arms is not exactly the same. It is slightly shorter on the left side than on the right (mouseover, green line). As a result, the square around the head (one quarter of the man) has moved slightly to the left overall. The line at the left elbow is also slightly shifted to the left side
- the left and right sides of the surrounding square tilt upwards by about 0.3° to the center (mouseover, red lines). This makes the surrounding square look as if it is slightly tilted backwards or forwards in perspective. This cannot be an error of the digital copy, because the surrounding circle is in contrast exactly round.
Why Leonardo does not draw these lines as precisely as the others is unknown.
III Golden ratio
In the accompanying text, Leonardo does not mention the golden ratio, although it is an important insight from the drawing.
The vertical Golden Ratio
- the center of the surrounding circle lies very exactly on the height of the golden section of the square, thus in the navel (blue point)
Furthermore, by steadily dividing the height of the body according to the golden section, it can be determined that the proportions of the human being can be depicted with the golden section (mouseover).
The horizontal golden section
The golden section can be shown at the Vitruvian man not only in the body height, but also in the body width.
- the armpit lies in the golden section of the body width with outstretched arms (dark blue line at right shoulder or upper horizontal line at navel)
- if a cubit is divided according to the golden section, the golden section marks the transition of arm and hand (right blue and orange area)
- the length of a thumb divides the length of the hand also in the golden section (right and left, blue/orange line)
It is unclear to what extent it was Leonardo's intention to show the golden section, since he always hits his marks only approximately, more or less precisely. Only the navel is really positioned exactly in the golden section of the body height.
Since the left shoulder is slightly shifted to the left (green line), the shoulder line there is not in the golden section of the width of the surrounding square (lower horizontal at the belly button). In addition, the transition of hand and arm on the left side is divided only very approximately in the golden section (black/white area). However, the length of the thumb here is in the golden section of the length of the hand, as on the right side.
It can be concluded that Leonardo, if anything, established the golden section on the right side with greater accuracy than on the left side. Forearms and the area between the armpits are the same length according to the geometric scheme: 1 cubit on the scale.
IV The hands
The arms held horizontally are marked with longitudinal lines at the roots of the hands (black and white areas).
- "the whole hand is the tenth part of the man"
- "from the hairline to the bottom of the chin is one tenth of the height of a man".
From this it follows that a hand is as long as the face is high (black and white lines on the face). If the width of two hands is transferred to the lower scale, it is exactly 5/6 of a cubit.
It shows now that the longitudinal lines at the hands did not refer to the golden section of the forearm, but to the length of the hands, because here the length is now exactly the same, since not from the elbow but from the edge of the square is measured.
However, Leonardo again deviates from the descriptive text. He writes the hands are "the tenth part of the man", but he has drawn in the length of the hand with 10 of 96 finger widths, as can be read from the scale (black and white line correspond to a hand length, respectively that of the face).
So Leonardo has rounded when he spoke of the tenth part: 10 of 96 correspond to 10/96 = 9.6 (~10).
V body height squared
Just below the scale of the drawing, is this sentence, highlighted by centering:
- "The length of a man's outstretched arms is equal to his height".
The greatest width that a man can span with his arms is at the level of the middle fingers, because they are the longest fingers. Both middle fingers of the horizontal arms form a line at the top of the chest. The distance of this line upwards is also subject to a proportion
- "From the top of the chest to the top of the head is one-sixth of the height of a man".
The distance of this line to the top of the head has the same length as the width of the hip (white lines) and shows 4/6 of a cubit on the lower scale.
VI Rotation points of the arms
[Translate to english:]
At the same height of the line spanned by the middle fingers, there is a line below the chin (blue horizontal line). This is marked by two endpoints (blue dots on horizontal line). The length of the line on the scale is 3/6 of a cubit.
The endpoints could show the rotation point of the arms, but would then be anatomically wrong. Arms rotate in the case at the shoulder joint, but for this the points are too far inside. Leonardo had to know this because of his anatomical studies and therefore probably did not draw rotation points.
- From the two end points of the blue horizontal line, two lines lead in a 22.5° angle to the middle fingers of the raised arms
- from the center of the square again a 22.5° angle leads down to the left shoulder (left white line)
- a 72° angle from the right shoulder upwards
The 22.5° angle is created by dividing a 45° angle, which in turn is created by dividing a 90° angle. 90°, 45° and 22.5° are the midpoint angles of a regular 16-, 8- and 4-corner (i.e. a square).
The 72° angle is the center angle of a regular 5-corner, which can only be constructed knowing the golden ratio.
Golden section and right angle or their direct variations have a high symbolic value in classical geometry. Therefore, it is not accidental that Leonardo emphasizes precisely these angles around the head.
In the construction of his paintings Leonardo almost exclusively combined the golden section (72°), square (90° angle) and equilateral triangle (60°) or their direct variations, so also here. The 60° angle is still missing, i.e. the interior angle of an equilateral triangle. According to the accompanying text, this is supposed to span between the legs of the man, but the angles deviate by 5° (I + mouseover).
Supposed disharmony of geometric relations is typical for Leonardo's art, and serves to create tension when tracing his works. In this way, the paintings lead from one realization to the next, until a final harmonious image emerges as the sum of all parts. It is therefore obvious to assume that this work also conceals another previously unknown level.
VII The face
- "The distance from the bottom of the chin to the nose and from the hairline to the eyebrows is the same in each case and, like the ear, is one-third of the face"
The face can therefore be divided into thirds (white area). Between the eyebrows and the tip of the nose are the ears.
A third of the face, like the sevenths above, cannot be transferred to the scale without a remainder: The face has the length of 1/10 of the body height of the man = 10/96, or 10 finger widths (smallest unit of the scale). If the 10 finger widths are now to be divided by three, one finger width would remain as remainder. It is exactly the width by which the surrounding square is tapered upwards.
The squaring of the circle
The possibility of squaring the circle, that is, constructing a square from a circle of the same area, has been a frequently studied topic in geometry since ancient times. According to the classical Euclidean doctrine this could only be done with the help of an unmarked ruler and a compass. There is no known case in which this was successful. It came thereby only to approximation constructions. In 1882 the mathematician Ferdinand von Lindemann proved the impossibility of this construction.
Is the Vitruvian man a hidden construction sketch for the squaring of the circle?
Numerous notes and geometric drawings prove that Leonardo was concerned with the squaring of the circle. The quotation at the beginning of this page is only one example of many.
The reason for the assumption that Leonardo would have shown in the Vitruvian man an approximate construction for a squaring of the circle, probably lies in his often quoted joy in creating riddles and hidden messages, which can be found among other things also in his paintings. And probably because of the prominent importance of the circle and square on this study sheet.
According to this theory, Leonardo proposed the squaring of the circle as follows.
The sheet was first engraved in 1784 by Carlo Giuseppe Gerli and published in the work "Disegni di Leonardo da Vinci". Since then, although it was accessible to the general public, it was known only in the art-academic environment.
Study object for artists
Painters in particular used the sheet to learn to paint people by memory. For if all people are based on the same proportions and differ only in their deviation from them, it is sufficient to remember only the deviation from the Vitruvian ideal, for example, the slightly wider nose, slightly longer arms or slightly shorter legs. A drawing created in this way, based on generally accepted proportions, will show great similarity to the original, even if the figure was not a direct model. This is what Leonardo suggests in his "Writings on Painting" and explains using the example of noses. Presumably, the facial features of the figures in his painting "The Last Supper" were created using this method.
The rediscovery of Leonardo in the modern age
At the beginning of the 20th century, the popularization of the complete works of Leonardo da Vinci began. The cause was initially the media sensation caused by the theft of the Mona Lisa in 1911. Subsequently, the public began to perceive Leonardo not only as a painter, but also became more and more interested in the scientific-technical aspects of Leonardo's work.
Of international significance in this context was an international exhibition on the universal genius Leonardo da Vinci organized in 1939 by fascist Italy under dictator Benito Mussolini, in which Leonardo's engineering work in particular was placed in the spotlight. Mussolini was keen to highlight the Italian Leonardo as the original inventor of the modern tank, the automobile, and the submarine, as well as his ideas on the use of ballistic weapons.
The Vitruvian Man was also exhibited there and subsequently gained increasing public reception because of its simple and clear formal language, style reduced to the essentials, and universal claim to represent a seemingly eternal aesthetic principle.
Le Corbusier's Modulor
The proportion studies of the Vitruvian Man were taken up in the 1940s by the important Bauhaus architect Le Corbusier. Based on this, the latter created his own system of proportions, the Modulor, which he used in most of his buildings.
Today, the Vitruvian Man can be found symbolically on every German health insurance card, the Italian 1 Euro coins and was or is a symbol for numerous scientific projects, for example NASA's Skylab 3 mission.
Website of the exhibiting museum: Galleria dell’ Accademia, Venice
Frank Zöllner, Leonardo, Taschen (2019)
Martin Kemp, Leonardo, C.H. Beck (2008)
Charles Niccholl, Leonardo da Vinci: Die Biographie, Fischer (2019)
Johannes Itten, Bildanalysen, Ravensburger (1988)
Frank Zöllner/ Johannes Nathan, Leonardo da Vinci - Sämtliche Zeichnungen, Taschen (2019)
Marianne Schneider, Das große Leonardo Buch – Sein Leben und Werk in Zeugnissen, Selbstzeugnissen und Dokumenten, Schirmer/ Mosel (2019)
Leonardo da Vinci, Schriften zur Malerei und sämtliche Gemälde, Schirmer/ Mosel (2011)
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