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In his writings on proportions in the treatise, De Architectura, Marcus Vitruvius Pollio emphasised the importance of the human body within its space, stressing further how townplanning should follow the octagon found in the geometry. These ideas were later encoded in the
Vitruvian Man by Leonardo da Vinci.
Here is a brief outlay by Vitruvius...
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De Architectura : The Planning of Temples
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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.
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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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Leonardo da Vinci Vitruvian Man
1513 |
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The passage, above, from Vitruvius' De Architectura, is a source of inspiration for Leonardo's drawing in his sketchbook, the famous Vitruvian Man. This passage also provides the Key of the composition of the ancient architecture, firmly revived in the Renaissance. The emphasis is on rationalisation of geometry, by means of small whole numbers to build the composition – in this manner Vitruvius delineates the proportion of a male human body that Leonardo so faithfully reproduced. Rationalisation as the basic rule of composition and the simplicity of an ingenious design (proportioned by means of small whole numbers) are often ignored.
Both Leonardo and Vitruvius make no mention of the Phallos as the center of the square, but it plays an important role. 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 5ipalms).
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Comparative measures derived from square and circle
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cubit
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foot
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palm
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inch
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finger
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4.00
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6.00
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24.00
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72.00
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9.60
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Diameter of square
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5.66
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8.48
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33.94
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101.80
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135.80
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| Diameter of circle |
4.83
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7.25
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29.00
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87.00
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116.00
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| Circumference of square |
16.00
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24.00
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96.00
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288.00
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384.00
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| Circumference of circle |
15.20
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22.86
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91.10
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273.30
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364.40
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| Area of square |
16.00
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36.00
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576.00
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5184.00
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9216.00
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| Area of circle |
18.35
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41.28
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660.52
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5944.70
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10568.30
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The relationship
between square
and circle |
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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 (2 x 12) 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 centres, the navel and Phallus, is according to rationalised 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 octagon 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 a dome placed on an octagram, and the gematric value of Greek spelling for Jesus, Ihsous = 888, associating the number 8 (octagram) with Jesus Christ.
Most importantly 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.
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The relationship
between square
and octagon |
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Further on Vitruvius 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, Pneuma = 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 1 x 2 x 3 x 4 = 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... (1 x 2 x 3 x 4) + (1 + 2 + 3 + 4) = 34, making it 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 & 22/7 |
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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 3 x 7 squares. 22 and 7 are connected through the rational approximation of p = 22/7.
Number 22 also invokes the total number of Hebrew alefbeth, and numbers 3, 7, and 12 (= 22 – 3 – 7) are the numbers of divisions of Hebrew alefbeth into three groups 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. 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 8 x 4 = 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 centre of the human body. The circle on Leonardo's drawing has it's centre in the navel, but if the figure is 'squared' the centre becomes the phallus which is, concerning this drawing from the compositional point of view, more important, since it is the centre 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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The progression
of squares |
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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 magic in terms of conflicting or opposing discourses as is the case today.
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* These quotations are from F. Granger's translation, Loeb Classical Library, 1970
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And here's Arnold Roth's take on Leonardo auditioning models for the Vitruvian Man...
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