Previous studies of proportion in Leonardo’s work have focused on the
drawings of human anatomy and horses, and on the Sforza and Trivulzio equestrian
monuments. Rarely has the interest of scholars concentrated on architectural
proportions (

With the publication of the first anthology of Leonardo da Vinci, edited by Jean Paul
Richter (

Inspired by the opportune appearances of the first three editions of

Reprinted from Scholfield,

Reprinted from Pedretti,

A systematic study of the architectural drawings of Leonardo was undertaken by Jean
Guillaume (

As a first approach to the study of the proportions in the architectural drawings of
Leonardo, I have focused only on the sacred buildings sketched in Ms. B.

These drawings are all similar subjects (mostly variations on the theme of
the central plan or of composites thereof);

they are all proposed according to the same mode of representation (usually a plan accompanied by a perspective elevation);

they were executed around the same time.

The subject of sacred buildings is perhaps the most striking of all those treated in
Ms. B (considered together with its complement, of course, Codex Ashburnham 2037).
Available for consideration are eighty drawings,

The drawings in question that depict central-plan churches are seventy-two in number. Among them are the planimetric schemes of Brunelleschi’s Rotonda degli Angeli (f. 11v) and Santa Maria in Pertica, in Pavia (f. 55r). In some cases these drawings consist of elementary graphic lines: ff. 15r (three plans in the right margin), 21r (two plans with corresponding elevations), 52r (a basilica plan at lower left), 93v (two small plans at upper right and left), and others showing more complex schemes.

A limited group of figures consists of more elaborate drawings of larger size. They
are drawn in pen and shaded for enhanced three-dimensional rendering that makes them
stand out from the sheets. This group is composed of pairs, each consisting of a
plan and a perspective exterior view, and each relating to a particular building
(ff. 17v-18r, 18v-19r, 21r, 21v, 22r, 24r, 25v, 39v, 52r, 93v, 94r, 95v).

The plans are always in the form of geometrical ‘wire frame’ diagrams.
Only at a later time did Leonardo partially or fully ‘dress’ them with
wall thicknesses: thicknesses that not only generate inconsistencies in the
relationships between solids and voids, in the structural plausibility of wall
thicknesses, and in the connections between the parts, but also change the
proportional relationships.

Elevations, on the other hand, are in the forms of perspective views: curves, lateral
surfaces of hollow solids, and shells that suggest that the volumes and shapes,
regardless of how all the internal elements of the plan can be developed
structurally to become articulated above ground. The exterior is only one possible
form through which the plan can evolve in space, as a kind of vertical extrusion.
The structure of the interior, in fact, is never indicated in section (except in the
case of f. 4r of the Codex Ashburnham 2037 = Ms. B, f. 94r)

With the composite plans, the perspective views always look toward the apse, i.e., the part that corresponds with the main central space of the building, which is the most complex and articulated. Clearly Leonardo is here reflecting on the basilical form in light of the central scheme.

Some plans, if not all, study the various possibilities of a given form, and are akin
to ‘ludo geometrico’ — ‘geometric play’ (

The multiple graphic solutions that Leonardo proposes in the pages of Ms. B are for his own use, just like the notes that sometimes accompany them and that integrate or explain some characteristics or novelties of the planimetric and spatial invention.

This study begins with the geometrical interpretation of the plans. It examines the
proportional relationships that connect plans with elevations in a few significant
examples. This interpretative approach only considers the intrinsic characteristics
of the drawings and does not introduce external elements into the analysis that
would attempt to transform drawings that are, and remain, only schemes and general
studies into something possible and concrete, or feasible. The investigation does
take the context provided by the content of the codex into account, for such context
reveals a certain manner of working on Leonardo’s part, through chains of
association. Indeed, it is no coincidence that Ms. B contains no fewer than fourteen
geometrical constructions involving plane figures, especially triangles and regular
polygons (ff. 12v, 13r, 13v, 14r, 17r, 27v, 28r, 29r, 40r), and that four of them
involve or include the octagon (ff. 12v, 17r, 40r), drawn beginning from a given
square, or from the circumscribed circumference thereof, or from an indicated side.
Numerous studies of buildings in the codex also insist on the octagon and on the
ribbed dome surmounted by a lantern with an octagonal plan, demonstrating, moreover,
the extent to which Leonardo continues to reflect on the dome of Santa Maria del
Fiore, both with regard to its form and to the geometry associated with its
structural qualities.

The first operation performed on the designs concerns, therefore, their geometrical
bases (in particular of the plans), in order to verify their outlines, especially
when the drawings were executed freehand. Such is the case, for instance, in ff. 34v
and 35r, where two drawings, apparently geometrical exercises, refer to the plan of
‘Santa Maria in P(er)ticha da Pavia’ drawn on f. 55r of the same codex
(Figs.

At left and upper center: Leonardo, Ms B, f. 34v and graphic elaboration thereof; in upper right, Ms B, f. 55r, detail.

At left and upper center: Leonardo, Ms B, f. 35r and graphic elaboration thereof; in upper right: Ms B, f. 55r, detail.

Relating the proportions of the plan to those of the perspective elevation, even when freehand drawings are concerned, is permitted especially when important sides of the building in perspective are presented parallel to the perspective frame. This, for instance, is the case in the sheets of Ms. B considered here. The perspective views are almost cavalier axonometrics; the vanishing point is always to the left, and the horizon is very high, so as to establish a view from above that reveals very well the concatenated assemblage of volumes. I have refrained from relating plans and elevations when the latter were too small to furnish sufficiently detailed information. Significant results are obtained, instead, when the plan and the perspective elevation are on almost the same scale.

An example in which it seems preferable

At left and lower right: Leonardo, Ms B, f. 52r and graphic elaboration thereof; at upper right: Ms B, f. 57r, detail.

The following examples concern one basilical scheme and three centralized structures
in which both the square and the octagon appear, and the planimetry of which is more
complex than might at first seem.

It is possible to establish a precise relationship between the plan and the
elevation of the composite building of f. 24r (

Questo edifitio è abitato di sop(r)a e di sotto; di sop(r)a si va p(er)
li campanili e vassi su p(er) lo piano dove sono fondati i 4 tiburi, e detto
piano à j° parapecto din(n)ançi, e di detti tiburi nessuno ne
riessie in chiesa, ançi sono sep(er)rati i(n) tucto. (‘This
building is habitable both below and above; the way up is accessed by the

Leonardo, Ms B, f. 24r: graphic elaboration of the two drawings in the upper half of the sheet.

Leonardo conceives of the four ‘tiburi’, the small domed structures
corresponding with the corners of the rhomboid body of the church, which touch
and connect with four of the eight sides of the drum, with no visual relation to
the interior, as buttresses, similar to the ‘tribune morte’ of
Brunelleschi (

Comparing the scale of the elevation with that of the plan, it is possible to
verify how the elevation is constructed geometrically from the plan, and more
particularly from only certain elements of the central square. The height of the
first cornice surrounding the building on the top of the semi-cylindrical body
of the apse and the lateral tribunes equals two modules; the height of the
crowning cornice (the support for the balustrades), by contrast, equals the
width of the nave: it is therefore equivalent to √2. The remaining portion
of the square outline of the building, excluding the apse, in the end determines
the height of the drum.

The study of f. 39v (

At left and above right: Leonardo, Ms B, f. 39v and graphic elaboration thereof; lower right: Ms B, f. 25v, detail.

Notwithstanding the incongruities between the plan and elevation, and the
imperfections in the perspective view (the exedra on the right occupies the same
space as the bell tower, and is against a side of the drum, while that on the
left is entirely separated and distant from the drum), in this case, too, the
elevation can be identified on the basis of the plan. The lower level —
corresponding to the parallelepiped on which the drum rests — has a height
equal to the sum of twice the apotema (radius of the octagon measured to the
center of any side) of one of the octagonal satellites and the side of the
square space between two of those satellites. It is also equal to twice the side
of the square space between two of those satellites (yellow line, in Figure

In the first pair, ff. 19r-18v (Fig.

Leonardo, Ms B, details of ff. 19r (plan) and 18v (perspectival elevation): graphic elaborations.

The plan, an inscribed cross with an octagonal nucleus, has been drawn by Leonardo both with the help of tools (lines are clearly drawn with the ruler and the four half circles of the perimeter are executed with a compass) and in freehand (the niches of the four-lobed chapels and the four circumferences along the diagonals of the square), expanding an earlier, entirely freehand drawing, still clearly legible in the upper left quadrant of the plan.

Geometrically the plan is composed of a square divided into nine minor squares,
some of which are in turn further divided into sixteen smaller squares. The,
four-lobed chapels thus each have a nucleus of sixteen squares and are
articulated around the major square. Each of these chapels penetrates this major
square by one-quarter of the chapel’s width, about to touch the central
octagon. The circumference, only fully sketched out in the part that corresponds
with the external surface of the chapels, has a diameter that can be defined
according to its tangentiality with the internal octagon. This octagon, as can
be verified directly from measurements, is constructed on the basis of the
square whose side forms the golden section (Φ) with the side of the
perimeter square. This construction can also be found to be reflected in the one
Leonardo adopts on f. 26v of Codex K2 (= Cod. K, f. 74v).

The comparison of the geometry of the plan and the perspective elevation suggests that the height of the side of the main parallelepiped is exactly half the square that circumscribes the plan; that the height of the drum stands in a golden section relationship with the height of one of the sides of the parallelepiped body; and, finally, that the dome will be as high as the parallelepiped itself.

The golden section occurs also in the proportioning of the central structure
whose plan is on the upper right hand side of f. 18r and whose perspective
elevation is found below on f. 17v (

Leonardo, Ms B, details of ff. 18r (plan) and 17v (perspectival elevation): graphic elaborations.

The construction of the plan first involves the definition of the circular chapels on the basis of the tangent with the circumference circumscribed about the central octagon. As can be deduced approximately from the drawing, joining the center of the octagon with the extreme opposites of the diameters of the circumferences of the chapels should equal the double radius. On the same drawing by Leonardo, it can be seen that the diameter of the octagon has a golden section relationship with the side of the circumscribing square. In this way the satellite circumference tangent to the upper left angle is directly defined. Drawing the circumference passing through the center of this satellite, therefore, and within the circumscribing square, will define the midpoints of the other seven chapels of the crown.

The elevation is constructed so that the compact parallelepiped base will be as high as the plan is framed by one side of the square and the line (parallel to it) passing through the centers of two sides of the octagon. The total height of the bases and drums of the radial chapels equals half of the plan (a half side of the square). The drum corresponding with the octagonal nucleus, finally, goes from the line that in the plan indicated the top of the parallelepiped body (which coincides with the plane indicated by Leonardo as ABCD) and that which, on the opposite side, passes through the intersection of the ideal circumference which contains the midpoint of the radial chapels with the extensions of the two opposing sides of the octagon.

In this proportioning of the elevations, more than in the other elevations
considered hitherto, emerges a true and proper proportional scheme. In fact, as
becomes apparent from the last scheme, at lower right in Figure

Most of Leonardo’s architectural drawings still await examination with regard
to proportion. Still, the exploration conducted on the sheets of Ms. B — which
is only slightly antecedent to Leonardo’s studies of the problem of the
Tiburio of Milan Cathedral, in which he thus comes into contact with the realities
of construction

Leonardo rapidly derives the elevations from the plans;

this method is facilitated either by the fact that the plans are drawn before the elevation, or by the reciprocal position of the two drawings;

in the simplest examples the elevations are defined starting from a particular element of the plan by means of progressions (for the pair at the top of f. 21r the progression is 1, 2, 4, 8);

in the more complex cases the lines passing through quite particular portions of the plan define the heights and the cornices of the exterior (ff. 24r, 39v, 19r–18v, 18r–17v);

in some cases the golden section intervenes either in the definition of the plan (ff.19r–18v, 18r–17v) or in that of the elevations (ff. 18r–17v).

The recourse to the golden section is not odd because Leonardo, who applies a simple graphic method, seems to have been familiar with it since his Florentine years and at least since the beginning of the 1470s.

The preparation of Figures

Translated by Matthew A. Cohen and Maarten Delbeke

To Scholfield (

Pedretti (

For the drawings of Ms. B and the themes explored here see, in particular,
Heydenreich (

For the central plan, in addition to the fundamental Wittkower (

In arriving at this number I have also considered structures that are not ‘churches’ per se, but that contain a centralized or basilical plan: a plan and perspective section of the ‘Pavilion of the Duchess’s Garden’ (‘Padiglione del giardino della duchessa’; f. 12r) and the ‘preaching theater’ (‘teatro da p(re)dicare’; ff. 52r, 55r, 95r).

The drawings are distributed in the ten folded sheets that originally formed the codex as follows (the folded sheet 10 constitutes the present Codex Ashburnham 2037): folded sheets 1 (5), 2 (13), 3 (24), 4 (8), 6 (12), 10 (18).

In ff. 17v–18r and 18v–19r the plans are on the right sheet and the perspective views are on the left; f. 21r bears two distinct pairs of drawings; in f. 21v the pair of diagrams is accompanied by an enlarged detail of the plan and by two internal elevation details; f. 25v bears two pairs of drawings; f. 93v contains five pairs of diagrams, while in f. 94r the plan and perspective elevation are accompanied by a partial section. The other sheets (22r, 24r, 39v, 52r, 95v) each contain individual pairs.

Exceptions are the drawings of f. 21v, in which the elevation is flanked by a fully drawn plan (at left) and a partial but larger plan (at right).

This practice also applies to the drawing pair found on f. 12r (the
Duchess’s Pavilion, or,

This norm holds also for the pair in f. 15r (with a wood truss roof over a square plan).

This problem also occurs in the graphic transposition of steps in Alberti’s

It is a church with two levels set on a square base with an inscribed octagon. Only a section view of a portion of it is shown. See also note 7.

For the proportioning of the cupola of Santa Maria del Fiore and the geometric
positioning of the intermediate ribs (sixteen in total, located in pairs, side
by side, among the eight angle ribs), see D’Agata, Di Teodoro and Mancini
(

In the first (Fig.

Leonardo also examines the ‘teatro da predicare’ (preaching theater)
in f. 55r of Ms. B (‘theaters for saying mass’) and f. 95r of the
same codex (‘place for preaching’). The design of f. 55r, on the
other hand, executed in pen, shows a centralized scheme (a square with four
exedrae — one drawn in pencil — and ambulatories), but hints at,
with the extension of some pencil lines, a possible longitudinal body — a
solution that is taken up and developed in f. 35v. As it is well-known, the
‘preaching theater’ consists of an amphitheater-like structure (or
simply a centralized scheme), tiered, and provided with a high central pulpit.
It is plausible to assume that in conceiving of such an architecture Leonardo
was influenced by ‘parlagio’, a term used in Florence to refer to an
ancient public amphitheater, home, according to local historians, of the
‘parliament’. Villani (

The planimetric scheme of f. 52r derives from a modular grid in which the basic module (M) corresponds to a bay of the aisles and the tri-portico that wraps around the longitudinal main body. The nave is equal to modules, and the octagonal domed space, the center of the triconch organism based on the square, is defined by constructing inside the larger square (of side equal to 6M) a smaller square, the corners of which fall on the centers of the sides of the larger square. The irregular sides of the octagon are equal to 2 and to √2. The sides of the two squares are between them are in the ratio 2: √2. The building, excluding the three exedrae, is formed of the sum of two squares.

The drawings of f. 57r bear, in fact, the annotation, ‘A è Santo
Sepulcro di Milano di sop(r)a. B è la sua parte socto tera’ (‘A
is the upper church of Santo Sepulcro of Milan. B is the part
underground’). Cf. Guillaume (

Reflections on the

I have omitted the most elementary diagrams such as, for example, those of f. 21r: in the first pair (top left), the octagonal plan has four rectangular side chapels, each half a side deep (2: 1) and twice as high (2: 1). The octagonal drum on which stands a hemispherical ribbed dome is also twice as high as the side of the octagon. Overall, the structure is equal to four times the side of this polygon. In the second pair (lower right) the central octagon opens to four square chapels the side of which is equal to that of the octagon (1: 1). These chapels develop in height according to a cube surmounted by an octagonal drum surmounted by a cupola, the edge of which is half that of the cube (2: 1). Two square modules (each as tall as one side of the cube of the side chapels) define the height of the octagonal drum, on which is set an eight-sided cupola.

The discrepancy between the transverse dimension of the proportional scheme (in
blue in Figure

I refer here to the well-known graphical method for the construction of the
golden rectangle from the square whose side becomes a proportional mean. On
Leonardo’s use of the golden section, see Sinisgalli (

For the