“Proportion is not only to be found in numbers and measures, but also
in sounds, weights, intervals of time, and in every active force in
- God the Creator as Divine Geometer
- The proportions of the human body in the manner of Vitruvius (The Vitruvian Man) © Ministero per i Beni e le Attività Culturali, Polo Museale Veneziano
God's most perfect creation
The belief that mathematical law
governs everything in the universe goes back to before the time of
Plato. During medieval times, God was seen as a sort of divine
“geometer” who created everything on the basis of number and
All of Leonardo’s investigations regarding natural
phenomena were carried out with a firm belief in the mathematical
principles underlying all forms. One might even say that he “thought”
in terms of proportions – sound, light, the shapes and dimensions of
all living things, all were believed to be governed by “omnipresent
measure” or divine proportion.
Because man was seen as God’s
most perfect creation, harmonic proportions were also believed to
govern the human form. In his drawing of Vitruvian Man,
Leonardo illustrated Vitruvius’s principle that a well-built man with
hands and feet extended fits perfectly into a circle and a square.
had to manipulate things a little to make the principle work. The
circle is centred on the navel, but the centre of the square is
positioned lower. He also altered the measurements of the human body as
supplied by Vitruvius in his treatise and applied his own measurements
based on his study of the life model. Nonetheless, the image is now
accepted as a true representation of Vitruvius’ findings, and a
credible image of the ideal proportions of the human body.
- Codex Ashburnham Fol. 93v - Designs for a centrally planned church. Photo RMN - © René-Gabriel Ojéda
- Santa Maria della Croce, Cremona
An ideal building
According to Vitruvius, an ideal building
was one governed by the proportions of the human body. Alberti believed
that harmonious measure governs our very souls. We intuitively sense
when the building we are in is harmoniously proportioned according to
natural law. Renaissance architects from Brunelleschi to Bramante took
all of this on board, and the ideas manifested themselves particularly
in the form of designs for centrally planned churches, which they saw
as the epitome of divine proportion.
During the late 1480s in
Milan, Leonardo thought at length about churches with centralised
plans. Major projects under discussion at that time, including a new
Cathedral for Pavia and the choir of Santa Maria delle Grazie, may have
stimulated his interest in architecture.
All of his churches, such as those on Codex Ashburnham
Fol 93v are expressed as three-dimensional “organic” spatial volumes,
drawn from a bird’s eye view. Each is accompanied by a ground plan, as
if to illustrate the “roots” from which each has grown. The designs are
mainly derived from configurations of circles and the squares, the
perfect geometric forms. Most comprise of a central area surmounted by
a large dome with a small lantern, surrounded by chapels. This
configuration offered the greatest possibility for invention and
allowed Leonardo to achieve maximum unity and variety. It may have been
inspired by Brunelleschi’s work in Florence – the unfinished church of
Santa Maria degli Angeli and the dome and lantern of Florence
While Leonardo’s designs were never built, his
ideas regarding centrally planned religious buildings had a profound
effect on his friend the architect Donato Bramanate. Bramante’s
reconstruction of the east-end of Santa Maria delle Grazie in Milan
transformed the interior into a magnificent centralized space that
exudes a feeling for Euclidean geometry. The exterior massing of
geometric volumes is highly reminiscent of Leonardo’s designs in Ms.B.
His proposed design for the new Basilica of St. Peters, which was
originally based on a centralised plan also exhibits the same sense of
organic unity that pervades Leonardo’s architectural designs.
- Albrecht Dürer, Studies on the proportions of the female body, 1528, Woodcut
- A skull sectioned The Royal Collection © 2005, Her Majesty Queen Elizabeth II
Harmony and humanity
While Leonardo realized that every
figure is different, all were believed to be governed by the same
underlying principles of harmony that governed the universe. The
microcosm of the human body was mirrored in the macrocosm of the
universe and vice versa.
Renaissance theorists had long been
occupied by the search for human proportionality. Alberti, in his
treatise “On Statuary” described detailed measuring techniques for
determining the relations of each part of the human body to the other
and to the whole.
Judging from Leonardo’s surviving
manuscripts, his planned treatise “On the human body” although never
realised, would have gone well beyond any previous investigation of the
human body. In a remarkable series of Studies of the human skull,
dated c1489, Leonardo investigated the architecture of the human skull.
The main axes of the skull are sectioned just as some of the church
designs in his architectural drawings, in order to locate the position
of the human soul within. Leonardo locates it at the point where the
proportional axes of the skull intersect.
The lengths to which
Leonardo went in order to establish the universal nature of his theory
of proportion was unprecedented. He refined modules for the human body
at rest and in motion and for detailed parts of the body, as in his Study of the facial proportions of a man seen in profile.
He also investigated the proportions of women and children, and may
have even written a treatise on proportions, now lost, which is
mentioned in the Treatise on Painting.
- Last Supper © SPSAE, Milano
The science of painting
Proportion was a means towards the
rational and mathematically correct portrayal of nature that made
painting a science. Linear perspective, as the surest means of
achieving correct and natural proportion in painting, became the new
“science-art” of the 15th century.
Analysis of the perspective of the Last Supper
suggests that Leonardo applied a system of harmonic proportions found
in music to the construction of the picture’s architectural space. The
base of the painting is divided into 12 units and the ceiling into 6
parts along the top. The height from uppermost stucco molding to the
horizontal bottom line is half the width, equivalent to 6 units. The
width of the rear wall is one third of the whole, or 4 units. The
proportioning of the widths of the tapestries on the side-walls (which
have lost their design and now appear as blank rectangles) follows the
These proportions find an analogy in the
theory of music. 3:4 is the tonal interval of a fourth; 4:6 a fifth and
6:12 and octave, the only intervals considered harmonious by the
ancient Greeks. The fact that Leonardo made a note of a number sequence
of a similarly harmonic numbers on an early preparatory sketch for the
Last Supper now at Windsor would appear to confirm his intentions.
- Ms L Fol 79v-80r Photo RMN - © René-Gabriel Ojéda
According to Leonardo, the musician composes harmony by
the simultaneous conjunction of its “proportional parts”. The painter
“grades the things before the eye as music grades the sounds that meet
the ear”. On Ms. L. Fol 79v. Leonardo investigates sound and tries to
establish strict mathematical proportions between the loudness of a
sound at its origin and its range, the point in space up to which it
can be heard. This “pyramidal law” was seen as a universal law of
proportion that could be applied to other natural phenomena such as
light, perspective and even mechanics.
law of proportional diminution did not always lead to fruitful
conclusions. In the absence of knowledge of advanced mechanics, he
attempted to apply it to the investigation of gravity, noting that
every falling object acquires increments of impetus. He was obviously
unaware of the Merton “mean speed theorem” which states that a body
traveling at constant velocity will cover the same distance in the same
time as an accelerated body if its velocity is half the final speed of
the accelerated body.
Undeterred, Leonardo investigated the
actions of levers, pulleys and balances at length according to medieval
proportional laws, illustrating endless demonstrations of harmonic
equilibrium. While most of the results had no practical purpose,
Leonardo seems to have taken delight in demonstrating the certainty of
proportional law for its own sake. Perhaps it confirmed his view of the
inter-connectedness of all things in nature, which he so often
disproved in the course of testing theory with experiment.