Holbein's Mastery of the Elliptical Construction










Holbein’s Sketch of the Holy Family with St. Anne and St. Joachim (151819)


ans Holbein the Younger employed two distinct painting styles. One was the staid,
formal style of the portraits for which he is widely known. They are devoutly
puritan in the black garments, stony expressions and unadventurous backgrounds,
with only a couple of exceptions. The other is an exhilarating drawing style
with a bold use of white highlighting and strong oblique perspective construction.
Oblique perspective was virtually unknown in Holbein’s time. Its only
valid exemplar is a curious painting of unknown attribution mentioned by the
historian Vasari. Raphael used a type of oblique perspective in his bold ‘Coronation
of Charlemagne’, in the Vatican in Rome, but analysis reveals that this
painting was constructed entirely intuitively, with no adherence to a unifying
perspective scheme.
Holbein’s sketch of the Holy Family with two saints therefore
offers a striking advance from the stolid onepoint perspectives of the early
Renaissance. It is staged in an unabashed oblique view, with the drama of the
eye level at the foot of the tableau. Rather than the plain archways of prior
work, Holbein develops a complex structure of receding vaults and chamfers.
To do so requires an understanding of the projection of the semicircular structures
in the portico to oblique ellipses in the plane of the picture. This is a challenging
geometrical construction, especially in the early 1500s. Even now, few artists
would know how to generate the nested ellipses in the right configuration to
match the intended structure.
In view of the timing of Holbein’s work, it is interesting
that Jean Pélérin (the ‘’Viator’) had recently
published an analysis of the twopoint construction in 1505. He was French,
and published in Toul, a town in the Lorraine district northwest of Switzerland.
The book was pirated in an illustrated edition Nuremberg, again just north of
Switzerland, in 1509. It is thus highly plausible that a copy was available
to Holbein in Basle at that time. His friend, Nicolas Copernicus, had lived
in the same area, moving from Padua to to Frauenburg, in 1512 and published
his ‘De Revolutionibus’ in Nuremberg in 1543.
Detailed analysis of the archway in this sketch leads to the
conclusion that Holbein used a sophisticated construction method for the elliptical
curves and the converging details, with a few minor conceptual lapses. The lapses
are interesting because they indicate that he must have employed a geometric
construction of some kind rather than an optical projection method, such as
a camera obscura. Use of an optical projection method might have resulted in
inaccuracies of transcription as he traced the image onto the paper, but it
would not have generated conceptual errors in the construction logic.

Fig. 1. Elliptical construction of
the portico


Holbein’s construction is sophisticated because the primary
construction of the portico contains a set of six nested circles, which project
to ellipses in the oblique view that Holbein chose for this sketch. At this
point in history, it is not clear whether artists appreciated the rule that
circular structures always project to ellipses in oblique view. The mathematical
proof is difficult and the result counterintuitive. Up until this time, artists
had mostly restricted themselves to circular arches shown frontal to the canvas,
or noncircular arches. Where oblique arches are shown, they usually are in
a narrow angle of projection where the details of the ellipse are hard to discern.
It is interesting, therefore, to attempt to identify the first use of accurately
elliptical construction.
Nevertheless, the construction analysis for this sketch makes
clear that Holbein’s curves are essentially perfect ellipses. For each
curve of the portico, an ellipse can be found that follows almost perfectly
the curve that he has drawn. It seems, then, that he must have been aware of
the fact that circles project to ellipses, and also have had access to some
method of drawing ellipses where he wished to place them. Not only this, but
the method must have allowed flexible control of the ellipse placement, because
his ellipses are all at exactly the same angle (30º in this case) and are
nested a fashion appropriate to the spatial relationships of the circular arcs
from which they derive. The angle of view is such that the lower left edges
of the contours within the funnelshaped portico all coincide. The outer circle,
however, shows an expansion in the plane of the wall. For this reason it is
larger at all points than the next circle for the inner rim of the coping, and
hence the lower edges of the two outer circles do not coincide. Finally, there
is a tubular section leading to the inmost curve, so again the lower edges should
not coincide.
Another place where the construction is demanding is in the
scallop shell fountain below the portico. Here the actual shape of the shell
alcove is a flattened ellipse; at least, that is how we perceive it when viewing
the sketch. The perspective foreshortening squeezes this ellipse and, by the
inverse geometrical theorem, returns it to a circular shape on the page. Apparently
Holbein was aware of this inverse property because the outline of the shell
forms an almost perfect circular arc.
Holbein has captured the elliptical projection of all these properties essentially
perfectly. These days, such a construction could be readily achieved by use
of a sheet of ellipse templates, but it is far from obvious what method Holbein
could have used to generate such a precise alignment. The only place where he
deviates from the required construction is in the two small circles at the corners
of the structure. The enclosing turquoise ellipse shows the required orientation
of the projection, at 30º to match the orientation of the other ellipses.
But it is very clear Holbein must have drawn this small ellipse by hand, since
it is quite wobbly. Moreover, the yellow ellipse indicates the bestfitting
orientation, which is at 20º rather than 30º. The mismatch is fairly
noticeable in the original.