The ratio's in the drawing

Leonardo da Vinci added a line with proportions under the drawing, in which we can see what proportions he used (within the red rectangle): (see fig. 1)

fig. 1

We can copy this line with the little vertical markings as seen below: (see fig. 2). (I did not draw all the lines on the left side, but they are equal to the lines on the right side.) Let us see whether there are ratios of 1:1,6180339 in da Vinci's line.

   fig. 2

Below is a line with the "real" Golden Section of 1:1,6180339: the line b-c has the same ratio to the
line a-b as the line a-b has to the line a-c (or the shorter line has the same ratio to the longer line as the longer line has to the whole line): (see
fig. 3)

                                                                                       fig. 3

At first sight, without calculating, we can see that this is not the case with the line made by da Vinci (see fig. 2). In his line the line a-b has the ratio to the line b-d (the ratio of the shorter line tot the longer line) of 1: 2,97. If we look at the ratio of the longer line to the whole line (b-d to a-d) we get a ratio of 1:1,36, a completely different ratio than 1: 2,97.

Of course we can now look at the ratio of the distance between the lines with the smallest distance (blue lines in fig. 2) and the distance between the lines with a little more distance (red lines in fig. 2), as mentioned in the website in which the ratio between the distance from the fingertip and the wrist to the wrist and the elbow was allegedly 1:1,618. (We have to take the average of the distances between these lines, because da Vinci drew this lines apparently by hand, so the distances differ a little.) This ratio is 1: 2,66. Again here is no ratio of 1:1,618.

Luckily, da Vinci made notes above and beneath the drawing, so we can see whether these will give more information concerning the used proportions

In fig. 4 we can see what da Vinci wrote beneath the drawing. For those who cannot read his notes or cannot read Dutch it says: (sentence with large letters directly under the line with markings): "The length of the spread arms of a man equals his height."
And beneath the notes:

"From the roots of the hair till the bottom of the chin is one tenth of the total length;
from the bottom of the chin till the crown it is one eighth of the total; from the top of
the chest till the roots of the hair it is one seventh of the total man. From the nipples
till the crown it is one fourth of a man. From the elbow till the top of the middle finger it
is one fifth; and from the elbow till the armpit it is one eighth of the height of a man.
The whole hand is one tenth of the man, the beginning of the penis marks the center of
the man. The foot is one seventh part. From the sole of the foot till just under the knee
it is one fourth. From just under the knee till where the penis begins is one fourth of a man."

                                                                                  fig. 4

The notes (which are above the drawing of the Vitruvian man) in fig. 5:

"The architect Vitruvius says in his work about architecture that the measures of
the human body are divided by nature as follows: 4 fingers constitute one hand width,
4 widths of a hand constitute one foot, 6 widths of a hand constitute one fore arm or
yard, 4 yards constitute the height of a man. And 4 yards constitute a pace en 24
widths of a hand a man's height; and these measures he used for constructing buildings.
One have to know that when spreading one's legs the height of the body decreases by
1/4 and when lifting and spreading the arms, the middle fingers are at the level of the
crown, the centre of the spreaded arms are at the navel and the space between the legs
is a equilateral triangle."

                                                                                    fig. 5

So far, everything is rather simple and straightforward. Now we're getting to the difficult part.
When we measure the distance from the top of the head to the navel (a) and from the navel to the bottom of the feet (b), it seems that we are seeing here the Golden Section: "a" stands to "b" as "b" stands to the whole lenght of the body (c). But this could be the case with all human bodies, so I measured my own body and I found the same proportion. Also, not all human bodies have the navel on the same location, and it is difficult to get a precise measurement. Whereas the Golden Section is a very precise ratio.

Nevertheless, this is leading to a few new questions:

  • Since this proportion seems to be present in any human being, what is the connection with Leonardo da Vinci, and with his drawing of "The Vitruvian Man / Homo Quadratus" ?
  • Did da Vinci know anything about the Golden Section?
  • If he did, why is the Golden Section not mentioned anywhere is his work?
  • If he did, why is the navel not marked with a short line, like the lines in the arms and legs?
  • The navel is not a very important part of an adult human being. So what is the reason that so many people, who are believers in the Golden Section, see the Golden Section in the drawing (or the human body)?

Although da Vinci knew Pacioli and in fact worked with him during a few years, there is no evidence that Pacioli used the Golden Section in other fields than mathematics. Pacioli asked da Vinci to illustrate his work "The Divine Proportion", but there is no evidence that da Vinci used the Golden Section in his own work. So far as we can see (by studying all the available texts), the navel in the drawing of Homo Quadratus is just the middle of the circle and that is all. And so it is with many human bodies, but not with all human bodies.
The facts that the navel in the drawing is not very clear (there are three points), that the navel is not marked with a line, and that da Vinci does not mention the Golden Section anywhere in his work, seems to point to the vision that da Vinci indeed did not use the Golden Section in his work. As I said earlier, the mentioning of the "Divine Proportion" by da Vinci (and Plato), did have another meaning, pointing to a view that all things on earth had "divine proportions", or were created by God in a certain manner. By this, they did not mean the Golden Section, which is a very precise ratio, but they used this as a figure of speech.

Also, when we measure the proportions of the body in the drawing by da Vinci exactly, we can see that the navel does not lie on the exact spot where it should lie according to the Golden Ratio.
See fig. 6: the green line represents the Golden Ratio and the red line is the location of the navel and the centre of the circle.

                                                                                fig. 6

Also, we can see three points, in stead of one point, depicting the navel. When da Vinci would want to make a point using the Golden Section, it would be more logical, to draw one precise point on the right spot. But he did not. Besides, who needs three navels? One should be enough. I used the middle of the circle to place the red line. I could also have placed the red line on one of the "other navels", since many people state that "the navel" is on the spot, marking a point of the Golden Section.

Text by Vitruvius

When we see the text written by Vitruvius himself, the matter becomes more clear.

``In the human body the central point is naturally the navel. For if man be placed flat on his back with his hands and feet extended, and a pair of compasses centered at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square.´´


The most logical conclusion is that Leonardo da Vinci did not use the Golden Section anywhere in his work, and this includes the drawing of the "Vitruvian Man".
It is clear, when we see the drawing, and his connotations on the very same paper, that he used the ratio's of the architect Vitruvius, and that he also did not alter these; he just copied them.
In his other works, like the painting "Mona Lisa", the Golden Section is also not to be seen.
(But there is another method, which can be seen in his work, and that is the Diagonal Method. See the links page.)


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                              Behavioral  Sciences, Vol. 46(1), Winter 2010,
Falbo, Clement: The Golden Ratio - A Contrary Viewpoint, The College Mathematics Journal, Vol. 36, No. 2, March 2005.
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Hemenway, Priya: Divine Proportion, Phi in Art, Nature and Science, Sterling Publishing Company, New York.
Herz-Fischler, Roger: Proportions in the Architecture Curriculum, Nexus Network Journal, vol. 3, no.2, 2001.
Herz-Fischler, Roger: How To Find the "Golden Number" Without Really Trying, Carleton University, Ottawa.
Howell, Alice: The Web in the Sea, Jung, Sophia and the Geometry of the Soul, Quest Books, London, 1993.
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Letze, Otto: Leonardo da Vinci, Kunsthal Rotterdam, Rotterdam, 1996.
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Markowski, George: Misconceptions about the Golden Section,
Pickover, Clifford: The Math Book, Sterling Publishing Company, New York, 2009.
Schoot, Albert v.d.: De ontstelling van Pythagoras, Agora, Baarn, 1999.
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                       Barrington, MA; USA, 2001.
Suh, H.A.: Leonardo da Vinci Notities, Parragon Books, Bath, England, 2006.
Thomas, Brian: Geometry in Pictorial Composition, Oriel Press, Newcastle Upon Times, England, 1971

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