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Vitruvian Man |
by David Bowman
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| Leonardo da Vinci, Vitruvian Man, 1513, dim. 25 x 19.2 cm |
A passage from Roman architect Vitruvius (Marcus Vitruvius Pollio), describing the perfect human form in geometrical terms, was the source of inspiration for numerous renaissance artists. Only one of these, the incomparable Leonardo da Vinci, was successful in correctly illustrating the proportions outlined in Vitruvius' work De Architectura, and the result went on to become the most recognized drawings in the world, and came to represent the standard of human physical beauty.
It was the version produced by Leonardo da Vinci, whose vast knowledge of both anatomy and geometry made him uniquely suited to the task.
De Architectura established a system of ratios for the construction of 'perfect' buildings that exhibited Vitruvius' three necessary principles of durability, usefulness, and beauty. Vitruvius' s system proved a durable one, helping to spark a renewal of architectural interest in the renaissance.
As Vitruvius believed that good architecture was in essence a continuation of the laws of nature, he demonstrated that his system applied to the formation of natural structures, including man.
Vitruvius, De Architectura: THE PLANNING OF TEMPLES, Book 3, Chapter I
1. The planning of temples depends upon symmetry: and the method of this architects must diligently apprehend. It arises from proportion (which in Greek is called analogia). Proportion consists in taking a fixed module, in each case, both for the parts of a building and for the whole, by which the method of symmetry is put to practice. For without symmetry and proportion no temple can have a regular plan; that is, it must have an exact proportion worked out after the fashion of the members of a finely-shaped human body.
2. For Nature has so planned the human body that the face from the chin to the top of the forehead and the roots of the hair is a tenth part; also the palm of the hand from the wrist to the top of the middle finger is as much; the head from the chin to the crown, an eighth part; from the top of the breast with the bottom of the neck to the roots of the hair, a sixth part; from the middle of the breast to the crown, a fourth part; a third part of the height of the face is from the bottom of the chin to the bottom of the nostrils; the nose from the bottom of the nostrils to the line between the brows, as much; from that line to the roots of the hair, the forehead is given as the third part. The foot is a sixth of the height of the body; the cubit a quarter, the breast also a quarter. The other limbs also have their own proportionate measurements. And by using these, ancient painters and famous sculptors have attained great and unbounded distinction.
3. In like fashion the members of temples ought to have dimensions of their several parts answering suitably to the general sum of their whole magnitude. Now the navel is naturally the exact centre of the body. For if a man lies on his back with hands and feet outspread, and the centre of a circle is placed on his navel, his figure and toes will be touched by the circumference. Also a square will be found described within the figure, in the same way as a round figure is produced. For if we measure from the sole of the foot to the top of the head, and apply the measure to the outstretched hands, the breadth will be found equal to the height, just like sites which are squared by rule.
4. Therefore if Nature has planned the human body so that the members correspond in their proportions to its complete configuration, the ancients seem to have had reason in determining that in the execution of their works they should observe an exact adjustment of the several members to the general pattern of the plan. Therefore, since in all their works they handed down orders, they did so especially in building temples, the excellences and the faults of which usually endure for ages.
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Neither Leonardo nor Vitruvius make no mention of the Phallus as the center of the square, but it plays an important role in Leonardo's success. The side of the square is 4 cubits or 24 palms as can also be measured on the ruler beneath the square, and deduced from the Vitruvius' text. The diagonal of such a square comes close to 34 palms (33.94... exactly). If we rotate the square around Phallos for 45 degrees we get the upper point of the circle, and from this we can calculate the diameter of the circle as 29 palms (the difference between the square and the circle is 5 palms).
| |
cubit |
foot |
palm |
inch |
finger |
| side of square |
4 |
6 |
24 |
72 |
96 |
| diameter of square |
5.66 |
8.48 |
33.94 |
101.8 |
135.8 |
| diameter of circle |
4.833 |
7.25 |
29 |
87 |
116 |
| circumference of square |
16 |
24 |
96 |
288 |
384 |
| circumference of circle |
15.2 |
22.8 |
91.1 |
273.3 |
364.4 |
| area of square |
16 |
36 |
576 |
5184 |
9216 |
| area of circle |
18.35 |
41.28 |
660.52 |
5944.7 |
10568.3 |
Table of comparative measures derived from square and circle
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| relation between square and circle |
The two squares, the drawn and the projection square, form an octagram Pell's series are the progressions that approximate the geometry of proportions originating from the octagram The first of Pell's series is:
1 - 2 - 5 - 12 - 29 - 70 - 169 ...
Both main numbers 24 (2x12) and 29 are members of this series. Leonardo used this progression to rationalise irrational geometry of square root of 2 that rules the geometry of the octagram. As a proof, all other numbers are derived as a combination of the members of the series. The distance between the two centers, the navel and Phallos, is according to rationalized geometry 2,5 palms, and it seems that the position of palms touching the square in Thau formation of the figure equals 9,5 palms, again the measure derived from rational approximation of the geometry of the octagram by means of Pell series. Also the diagonal of the square is calculated from the series as 2x(5+12) = 34 palms.
The composition founded in octagram is one of the most spread cannons of proportions in ancient and classical architecture clearly presented by enormous opus of research done by Prof. Tine Kurent of Ljubljana School of Architecture, Slovenia. Almost every important christian church has dome placed on an octagram, and gematric value of Greek spelling for Jesus, ΙΗΣΟΥΣ = 888 associates the number 8 (octagram) with Jesus Christ.
Is it really so simple?
According to the proposed rationalization of Vitruvian Man is the geometrical construction also very simple. The composition is based on a square, which is duplicated and rotated for 45º to form an octagram. The distance between the base line of the first square and the apex of the rotated one simply represents the diameter of the circle.
Even Vitruvius mentions the value of the octagon, though in a bit obfuscating manner, while describing the winds:
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4. Some have held that there are four winds: the Solanus from the equinoctial east, the Auster from the south, Favonius from the equinoctial west, and Septentrio from the north. But those who have inquired more diligently lay down that there are eight: especially indeed Andronicus of Cyrrha, who also, for an example, built at Athens an octagonal marble tower, and, on the several sides of the octagon, had representations of the winds carved opposite their several currents. - Book I. c. VI: 4
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Further on he describes that even the town-planning has to be done according to the octagonal scheme of the winds. It seems that Leonardo incorporated, in multiple layers, the whole cannon suggested by Vitruvius. It is a strange coincidence that the area of a square with a side of 24 palms is 576 square palms, which is gematric equivalent of Greek word for the wind meaning also Spirit, ΠΝΕΥΜΑ = 576.
The calculated area of the circle is 660 square palms, and the difference between the circle and the square is 660-576 = 84, which is invoking the gematric value of the master himself:
LEONARDO = 12+5+15+14+1+18+4+15 = 84.
The 'fundamental' measure of 24 palms comes directly from Vitruvius' definition of human height being 4 cubits or 24 palms. Number 4 is of great importance in the composition of the drawing, emphasized with a square. Number 24 is important because it is 1x2x3x4 = 24, comprising the multiplied members of the Tetraktys. Vitruvius postulates the smallest module the division of 10 which also resembles the Tetraktys along with the previously mentioned diagonal of the square, 34 palms long: (1x2x3x4)+(1+2+3+4) = 34 - the Tetraktys numbers multiplied and added up. Half way between 24 and 34 is the diameter of the circle, 29 palms: (24+34)/2 = 29.
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| module 8 and 22/7 |
The predominating module of the composition of human body is eight and many lines on the drawing coincide with the lines of the modular plain. The figure in the form of the letter Thau embraces 22 squares of the composition, whereas on each half remains 21 or 3x7 squares. 22 and 7 are connected through the rational approximation of π = 22/7.
Number 22 also invokes the total number of Hebrew alefbeth, and numbers 3, 7, and 12 are the numbers of divisions of Hebrew alefbeth into three groups of letters as described in Sepher Yetzirah - The Book of Formation, a fundamental kabbalistic treatise on the nature of letters and words, suspected to originate from 13th century but it could be a lot older. The 32 Paths of Wisdom described in Sepher Yetzirah comprise 10 Sephiroth and 22 Letters. 32 Paths are written in the circumference of the modular square 8x4 = 32. The figure is positioned in the shape of T(hau), the 22nd, last letter of alefbeth. The Tetraktys and the kabbalistic Tree of Life are comparable glyphs, both comprising the ten spheres.
Vitruvius is reporting that the navel is the exact center of the human body. The circle on Leonardo's drawing has it's center in the navel, but if the figure is 'squared' the center becomes the phallos which is, concerning this drawing from the compositional point of view, more important, since it is the center of the underlying geometry that outlines the basic features of the figure.
The most fundamental composition consists of a circle, a square, and a triangle, a sigillum known to magicians and alchemist, sometimes called the Universal Seal of Light or the Seal of Hermes. The compositional triangle on this drawing is concealed, even though that it outlines important segments. It is drawn in the circle within the square and it coincides with the progression of squares as depicted on the illustration.
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| progression of squares |
The main proportional lines come from the progression of squares, every second square is half the size of the original, and the measures thus obtained are the same as described by Vitruvius.
Distinguished is also the triangle with the size of a square and apex in the navel.
It seems that the drawing, or better the original design as explained by Vitruvius, contains many layers of geometry and symbolism that concord in one single image delineating the proportions of the human body. This idea of 'reason' governing 'form' was the fundamental theme of the Renaissance and is traceable in best architecture and art in general. It would not be odd if Leonardo had a close contact with scholars that spread the source of the Renaissance thought which didn't distinguish between art, science, and magick in terms of conflicting or opposing discourses as is the case today.
Bibliography:
Vitruvius, De Architectura, F. Granger's translation, Loeb Classical Library, 1970
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