Leonardo's Man

Poststructural Pub
1

The Vitruvian Man, Leonardo da Vinci:

This is an interesting work of art, as Leonardo interprets notes from the famous Roman architect Vitruvius Pollio about proportions in the human body. The drawing is surrounded with text written in mirror image in which Leonardo explains the proportions depicted.

It is widely believed that Leonardo used the drawing to express his view that Man in his bodily construction expresses the harmonies believed to exist in the wider cosmos. I am not really interested in this viewpoint, and so rather discuss this work of art outside the sphere of religious topics (i.e. IFC). We all know that human bodies come in many different shapes and proportions. They may point to some general proportions, which are statistically prevalent. If this is due to an “ideal” human body, a cosmic or divine influence, is not meant to be part of this discussion.*

What always fascinated me in Leonardo’s Man is the way the figure is put into a square and a circle: with outstretched arms and hands, the man is a wide as he is tall (you can check your own body to see how true this is for you, you may have shorter or longer arms than him). And at the centre of the square are the man’s private parts, the pubic bone is found halfway between soles of the feet and top of the head. On the other hand the superimposed man stretching legs wide and arms upwards, so the finger tips are aligned with the top of the head, is within a circle which has as its centre the navel. So we find two clear centres.

If I am not mistaken, it has been popularised through “The Da Vinci Code” that the dimensions of circle and square depicted are related: that the radius of the circle is in Golden Proportion to the length of the square. I don’t think it is. After downloading an image from the web I printed it and measured it. I found that the square is not exactly a square, and the circle not exactly circular, but a little bit elliptical. I cannot imagine that Leonardo did not draw a perfect square and circle, so I adjusted height and width of the image to achieve a perfect square and circle, then made another printout and used that to compare proportions in the image.

It became clear that the circle is not derived from the square by using the Golden Ratio of 1.618. Rather than that, Leonardo might have rotated the square around its own centre 45 degrees, so that a corner will be right at the top above the head, and drawn a circle around the navel point and through this top point, the top corner of the rotated square, or rather halved the length from this top point to the base line at the feet, to arrive at the centre, which became the navel point. This gives a clear numeric relationship of square and circle, and it has nothing to do with Phi, the golden ratio, but with the squareroot of 2, which is found in the diagonal of a square.

Did Leonardo use the golden ratio in his art? The golden ratio had been known since antiquity. The mathematician and friar Luca Pacioli wrote “De divina proportione” in 1496–98, about mathematical and artistic proportion, especially the mathematics of the golden ratio, and he taught Leonardo Da Vinci mathematics at that time. Leonardo drew illustrations of regular solids for the book, while he took mathematical lessons from Luca Pacioli. So he was introduced to the golden ratio at that time at least. But the Vitruvian Man was drawn around the year 1487, and the text does not mention the golden ratio.

I am curious if anyone has any insights into this, and the wider topic of proportions of the human body.

* added postscript:
Of course it could be discussed here, but I don’t want to discuss Leonardo’s Man under an assumption that the origin of the human body’s proportions are in god, divine, super natural or designed, even though Leonardo himself may have believed so. I rather talk about the visible facts.

2

There is quite a lot in the Wikipedia article.

I’ve read articles mentioning the golden ratio in Da Vinci’s architecture. He may have used it without being aware of the mathematics of it. Once you understand that the ratio 0.618(etc) to 1 is the same as the ratio 1:1.618(etc), you have it.

The trouble is, you can find the golden ratio in a lot of art and a lot of architecture, because it is a pleasing proportion. It doesn’t mean the artist or architect knew the mathematics.

Le Corbusier had a bee in his bonnet about human proportions as well. It’s a long time since I looked at this stuff…

3

Finding that the circle was actually an oval would cause me to wonder about the provenance and accuracy of the image I was measuring. Are you sure that somewhere along the way that image wasn’t reproduced slightly condensed along one axis?

4

Yes, I wondered about the accuracy of the digital image. I have no idea why the image shows this small discrepancy. It might have been condensed along one axis. I just assumed that Leonardo drew an exact square and a circular circle. With that assumption I then corrected the image before printing to show an accurate square. Just thinking a bit more on this: it could also be that the printer distorts the image. I need to measure the pixel width and height of the square to be more sure on this. When you click on the image I included from wikipedia, you get a large one. Worth checking that out.

5

It is very easy to distort an image along one axis in any image editor more complicated that MS Paint.

6

I just looked at the large image from wikipedia:
the sides of the square are not even at right angles, it is a bit skewed.
This may be due to the camera/camera technique used taken the original image.
Sides of square in image in pixels:
left = 1580, right = 1570, top = 1595, bottom = 1605.
The circle measures 1940px width by 1918px height.
If you stretch the image vertically by a factor of 1.015 you get a more precise square and circle, although the square is still skewed/distorted by about 8 pixels (0.5%).

7

Americans may not know that this image appears on the 1 Euro coin.

8

Only the Italian one, that is. Americans may not know that each country of the eurozone has different images for their coins.
Most countries have their sovereign/monarch on the coin. The republics have something more meaningful ( WA Mozart or an Irish harp) depicted on the coin.

9

I think now that I got the above wrong. The rotated point of the square comes very close to the circle’s topmost point, but not precisely, and there are no marks to suggest that Leonardo used such approach to construct the circle. On the other hand you can see a line at the base of the neck with two prominent marker points at each end. The base of the neck line is 4 palms below the top hair line (the top line of the square), as per Vitruvian’s notes, and it is 3 palms wide.

With a compass you can take the distance to the side of the square, at the horizontally outstretched finger tip, and find the point on the square’s topside with that distance from the marker point, basically rotating the arm upwards (using the compass from the marker point). From that point to the navel gives you the circle’s radius.

And the navel point is taken as two and a half palm width from the centre of the square (there is no note from Vitruvian for the placement of the navel, so it must have sprung from Leonardo’s own observations). Leonardo included a ruler with markings for the finger width, palm width, foot and cubit as per Vitruvian’s notes.

I think Leonardo followed Vitruvian’s notes quite closely, but worked out through his own physical observations and the mechanics of arm movement how the circle’s radius can be derived at. The golden ratio was not used for this, nor an imaginary square on its point.

One strange thing I found is that the text says “the length of a man’s foot is one-sixth of his height” or four palms, but the foot stretched to the right from the centre line in the image is clearly a lot less than 4 palms, barely 3 1/2 palms. It looks like seven of those feet would equal the man’s height, not six. I think Leonardo got his drawing right, and Vitruvian’s proportion of six feet to the height is just wrong. this again shows that Leonardo is more concerned in expressing proportions as observed, rather than follow a canonical rule.

10

I didn’t and I will get one now. Thank you.

Slightly off topic. I’ve seen TV shows about beauty and proportions and how those people who are perceived as most beautiful are most average in their facial proportions.