Koryncki kapitel z Hermopolis Magna (Egipt) i jego geometryczna analiza

2026

3(87)

Marek Barański*

The Corinthian capital of Hermopolis Magna (Egypt)

and its geometric analysis

DOI: 10.37190/arc260301

Published in open access. CC BY-NC license.

Abstract

Paper presents study on the Hellenistic Corinthian capital (240 BC) from Hermopolis Magna (Egypt) Vitruvian proportions of a capital had been

conrmed partly, thus another geometrical solution was analysed. Due to completed geometrical diagrams it became obvious design was rooted in

a system created by gures being inscribed into a bigger and a smaller circles equal to a lower and an upper diameters of the column.

Key words: Corinthian capital, Hermopolis Magna, ancient geometry

Introduction

Vitruvius recounts the genesis of the Corinthian capital

in romantic fashion in “De Architectura” (Vitruvius 1914,

IV, 1, 9, 10, 104–106). The great sculptor Callimachus, he

says, noticed a basket set upon a girl’s grave and protected

by a roof tile; acanthus leaves had sprung up around it, and

their tendrils curled under the tile. Vitruvius commented it

with a concise set of proportions that Greek builders were

thought to have adopted:

– the abacus slab should be 1/7 of the capital’s height;

– the capital’s total height equals the column’s lower

dia meter, while its calathus equals the upper diameter;

– dividing the calathus into three equal zones locates rst

the lower, then the upper row of acanthus leaves, leaving the

third zone for the volutes and the ornament between them;

– the diagonal of the square abacus should equal twice

the upper diameter of the column, and thus twice the mea-

sure of the calathus;

– each corner recess of the abacus is to be one-eighth

of its side.

Since the Renaissance hundreds Corinthian capitals had

been analysed following this description. They were carved

according to Vitruvian model with modication in a form

of leaves and volutes. Discovery of Hellenistic Corinthian

capitals brought a problem, they were not composed ac

cording to a perfect description of a Roman author. Still,

it is worth stressing that, although Vitruvius wrote at the

close of the 1

century BC, we possess no textual descrip-

tion – and thus no sure notion – of the appearance of earlier

Corinthian capitals. The discussion that began more than

200 years ago is still ongoing (Heilmeyer 1970; Robert-

son 1979, 140–162; Pollitt 1986, 247, 248; Lawrence 1996,

138–145; Wilson Jones 1989; 1991; 2015; 2022). It is note-

worthy, too, that the proportional canons Vitruvius cites for

Ionic columns rely on the treatise of Hermogenes in the 2

century BC. Yet Hermogenes’ work has not survived, nor

have the writings of one Arcesius/Argelius, who is said to

have discussed the Ionic and Corinthian orders in the same

century. We can only infer that the principles embodied in

Hellenistic capitals ca. 360 BC adapted the norms current at

the time – perhaps akin to, but possibly quite distinct from,

the later Vitruvian model.

There are few scientist who proposed explanation of

a problem and presented an origin of the earliest Greek Co-

rinthian capital as a period when form of capital had been

individually shaped by Hellenistic artists (Homolle 1916;

Ebeling 1924; Pedersen 1989; Scahill 2009). Scholars have

tried to account for this divergence by analysing the measure-

ments of individual capitals, hoping there by to detect further

underlying relationships (Bauer 1973). The discrepancy

* ORCID: 0000-0001-8092-9108. Professor emeritus, Poland,

e-mail: pkz.baranski@gmail.com

Marek Barański

itself was long taken as evidence of an early phase, a mo-

ment when the rules that Vitruvius would later codify were

only beginning to crystallise. Our recent study on Hellenis-

tic capital from Hermopolis Magna in Middle Egypt pro-

vide highly suggestive evidence that substantially revises

this long-standing perspective.

Corinthian capitals

from Hermopolis Magna

Excavations conducted in the 1940s by Alexandria Uni-

versity at Ashmunein (ancient Hermopolis Magna), on the

area called the “Agora”, produced a series of striking nds

(Wace 1959). Among them there were remnants of sixteen

uted columns with bases and Corinthian capitals of the

Hellenistic period (Wace 1946, 13, 14; Wace, Megaw and

Skeat 1959, 4–11, pl. 1, pl. 15.2, pl. 16). The group has been

dated to about 240 BC, when Greek katoikoi were settled

at Hermopolis Magna in Middle Egypt. They evidently set

out to raise a range of civic and sacred buildings whose

forms would faithfully reect the style of Greek architec-

ture (Wace 1959, 4–11; von Hesberg 1978; Pensa bene

1993, 3–18; Barański 1996; 2004; 2019, 8–10; McKenzie

2007; Hoepfner 2020). The architect at Hermopolis was

pro bably trained in Alexandria, as suggested by the very high

quality of the capitals’ carving. In general, these capitals

correspond to the Style Group I of Konstantin Ronczews-

ki’s typology of Corinthian capitals (Ronczewski 1927),

the group that includes some similar examples to earliest

ca pitals from the Epidauros Tholos, built in 360 BC.

The discovered Hermopolis capitals display crushed cor -

ner volutes; more importantly, traces of their original po -

ly chromy were discovered on the surfaces, greatly enhanc-

ing the signicance of the nd. One of the capitals was

taken to the Graeco-Roman Museum in Alexandria, where

it became a prized exhibit – though not prized enough to

prompt thorough documentation or analysis. It has been

reproduced time and again in surveys of Hellenistic art

and architecture, yet always without in-depth study (Adri-

ani 1972, 124, 125, tav. XV; Pensabene 1993, 324, tav.

8, 43–47; Bauer 1973, 117, 122, 123, taf. 32.5; von Hes-

berg 1978, 138, g. 131; McKenzie 2007, 56; Bassioni

2022, 46).

In 1987, while working in the Hermopolis – Ashmunein

Basilica as part of the Polish–Egyptian Mission, I under-

took a systematic study of this Hellenistic capital (Barański

1996, 104; 2004; 2019, 9, 10, gs. 1–4). The discovery of

its missing corner volute during excavation made an almost

complete reconstruction possible. I executed a documenta-

ry drawing at a scale of 1 : 5 that records the capital’s form

in full (Fig. 1a, b). It should be noted that in 2020 Wolfram

Hoepfner, in his study of the Hellenistic nds from Her-

mopolis, presented his own colour drawing of the capital,

derived from Wace’s description (Hoepfner 2020, 43–45,

abb. 33, 35, 36, 49, 51). Nevertheless, in that reconstruc-

tion the corner volutes are rendered too large, altering the

overall proportions of the piece, while a closer inspection

reveals several additional inaccuracies, suggesting that the

drawing was executed from photographs rather than from

direct measurement.

Fig. 1. Corinthian capital from Hermopolis Magna:

a) capital with discovered corner volute (photo by M. Barański), b) coloured reconstruction of a capital (drawing by M. Barański)

Il. 1. Koryncki kapitel z Hermopolis Magna:

a) kapitel ze znalezioną wolutą (fot. M. Barański), b) kolorowana rekonstrukcja kapitela (rys. M. Barański)

a b

The Corinthian capital of Hermopolis Magna (Egypt) and its geometric analysis

Revising the capital’s proportions

In the next phase of inquiry I analysed the proportions

of the reconstructed capital and found that only certain el-

ements conform to Vitruvius’ canon. Before moving on to

a proportional analysis, it was necessary to set out the cap-

ital’s overall dimensions. I found a problem when compare

dimension taken at the site with those published by Patricio

Pensabene, who recorded the museal capital – a total height

of 89 cm; a lower-leaf zone of 25.3 cm; an upper-leaf zone

of 39 cm; a calathus height of 78 cm; and an abacus height

of 11.4 cm (Pensabene 1993, 324). Pensabene also men-

tioned a scamillus diameter of 80 cm, but he did not supply

the lower column diameter, because only the capital – with-

out its base – had been transported to the museum. A simple

division of the capital’s height by that of the abacus yields

the unusual ratio of 1 : 6.80, which might be accepted as

plausible. Yet if one divides the overall height into seven

equal parts and the calathus into six, the resulting abacus

units dier (12.71 cm vs. 13 cm). Nor is the arithmetical

sum of Pensabene’s gures satisfactory: 78 cm + 11.4 cm

pro duces 89.4 cm, and not the stated 89 cm.

Subsequent measurements taken on site at Ashmunein

dier slightly. The calathus height is 77.4 cm and the aba-

cus proper 11.4 cm, giving a combined height of 88.8 cm.

Above the abacus a further 1.4 cm of stone was left as

a working margin, a ledge. In my rst analytical step, fol-

lowing Pensabene, I adopted a capital height of 88.8–89 cm,

an abacus height of 11.4 cm, and a corrected calathus

height of 77.4 cm; but the resulting abacus-to-capital ra-

tio of roughly 1 : 6.8–7 remained unsatisfactory (Barański

2019, 9). The capital’s height was therefore adjusted to its

true dimension of approximately 90–90.2 cm, a gure that

includes the 1.4 cm of surplus stone intentionally left on

the abacus. With this margin, a ledge – destined to bear the

architrave – the capital fullled a cardinal proportional rule

of the architectural order. The slight recess cut at the abacus

corners, forming this way a ledge was probably dictated

by the relative softness of the limestone: if the corners had

been fully nished, there was a real risk that they could

be damaged during mounting an architrave. Signicant-

ly, the corrected height also matches the measured lower

diameter of the column, which is likewise about 90 cm.

Once this dimension is incorporated into the abacus measu-

rement, the numerical discrepancies disappear and all g-

ures agree.

At this height the capital can be divided into seven equal

parts, yielding an abacus unit of 12.8–12.9 cm. That same

unit suits the calathus when it is divided into six parts, for

its measured height of 77.4 cm likewise equals six units of

12.88–12.9 cm. The working assumption, then, is that the

capital was laid out according to a single abacus module

common to both the capital as a whole and the calathus;

the unit should be identical in each case. Such a scheme

would corroborate Vitruvius’ partitioning. In consequence,

one-seventh of the abacus slab and the overall heights of

both the capital and the calathus correspond to the lower

and upper diameters of the column. This yields a set of

regular divisions in which normal workmanship tolerances

of roughly 3–5 mm may be allowed. Ultimately, the aba-

cus module may be reckoned either as 1/7 of the capital’s

height or as 1/6 of the calathus height.

In accordance with Vitruvius’ prescription, the width of

the abacus slab can be determined by treating its diagonal as

the equivalent of two upper diameters of the column shaft

– that is, twice the height of the calathus (Fig. 2a). On this

basis the abacus diagonal should measure 2 × 77.4 cm =

154.8 cm. Dividing this value by √

to obtain the side of

the square yields 109.45 cm, or approximately 109.5 cm.

As a control, the calculation may be repeated using the

lower diameter – namely, the capital’s total height – which

gives a diagonal of 2 × 90.2 cm = 180.4 cm and a result-

ing side of circa 127.7 cm. Although this second gure ap-

pears to approach ten abacus modules of 12.9 cm each, the

discrepancy is in fact too great to justify such a reading.

When the two alternative abacus sizes are plotted on the

reconstruction drawing, the corners derived from the upper-

dia meter scheme sit neatly, whereas those derived from the

lower-diameter scheme project well beyond the volutes. In

the discussion that follows this mismatch will not aect our

analysis of the decorative proportions, but we shall return to

the question of how the abacus dimension was established.

According to Vitruvius’ description, the calathus should

be divided arithmetically into three equal parts in order to

x the positions of the leaves and the volutes. In the Her-

mopolis capital, however, the subdivision is dierent. The

height of the upper leaves equals exactly one-half of the ca-

lathus height, while one-half of the entire capital height

marks a distinctive level situated above the upper leaves;

this line passes through the stalks of the inner owers and

through the collars of the middle stems.

A further test experiment was therefore made with

a smaller unit, dividing not into three but into six parts, yet

this scheme added little. A division into eight parts was also

tested – prompted by the number of corner volutes and of

leaf tiers – but that analysis likewise failed to yield a satis-

factory solution. It was observed, meanwhile, that a num-

ber of characteristic points on the capital are determined by

modular grids; still, identifying several key points did not

by itself explain the overall composition. One additional

result of these trials was the realisation that the divisions

employed in the capital could expand from a smaller to

a larger “circle”.

Even more surprising was the discovery that the relations

derived from the column diameters can be linked to the base

in the following way. The base height equals two modular

units (marked at drawings as a UV), so that when the base

and capital are combined the total comes to nine units of the

abacus measure. Remarkably, those nine units correspond

to the lower torus diameter of the base, while the upper to-

rus diameter equals eight such units. Thus four concentric

circles are introduced: “A” comprising seven units, “B” six

units, “C” eight units, and “D” nine units, all referred to the

abacus module [These circles would measure, respectively,

A = 90.2 cm, B = 77.4 cm, C = 103.2 cm, and D = 116.1 cm]

(Fig. 2b). Although such a direct linkage of capital and

base is never encountered in practice, it signals that additio -

nal dependencies may exist, inuencing the mutual rela-

tionships of the order’s components – dependencies that

ought perhaps to be taken into account in the analysis. These

Marek Barański

Fig. 2. Hermopolis capital: a) geometric reconstruction of an abacus size depending a lower (red) and upper (green) diameter of a column trunk.

Upper diameter (green) indicates a right solution, b) partition of capital and base

following section of circles A, B, C, D by 4 and 8. Vitruvian

partition unite (UV) of a measuring 1/7 of overal capital height and equal to abacus size

(elaborated by M. Barański, drawing by K. Jezierski)

Il. 2. Kapitel z Hermopolis: a) geometryczna rekonstrukcja wielkości abacusa w zależności od dolnej (czerwona) i górnej (zielona)

średnicy trzonu kolumny. Górna (zielona) średnica wskazuje właściwe rozwiązanie, b) podział kapitela i bazy według kół A, B, C, B mających

podział na 4 i 8. Witruwiańska jednostka podziału (UV) mająca wielkość 1/7 wysokości całego kapitela odpowiada wielkości abakusa

(oprac. M. Barański, rys. K. Jezierski)

inter

connections pertain not only to simple vertical subdivi-

sion but also to the horizontal layout, where the base dimen-

sions represent a modular enlargement of the lower column

diameter. Here, we should mention, a horizontal division of

circles by 4, 8, 12 provides a gene ral distribution of capital

volutes, and leaves of upper and lower row. While these

observations were helpful, they did not yet provide a clear

key to the arrangement of the capital’s decorative scheme.

Another concept of a modular grid adoption was also an-

alysed. The Egyptians employed precisely such a system in

their reliefs and architecture, enabling them to coordinate

dimensions and to position distinctive points within a sculp-

tural composition (Rossi 2007, 113–127). It is quite possible

that a comparable modular grid – conceived as a matrix or

template – was likewise used when carving the capital it-

self. Such a device would have allowed the sculptor, as the

work progressed, to verify the placement of critical points

on every face of the block, in both the vertical and horizon-

tal axes. Although no nished examples of stones so pre-

pared have survived, incised guidelines on Roman blocks

have been documented and analysed (Asgari 1988; Toma

2015). A comparable system of setting-out lines is attest-

ed on late-Hellenistic monuments in Nabataean kingdom,

whose builders were inuenced by Ptolemaic architects

(Dentzer-Feydy 1995). Scribed lines preserved on the upper

and lower faces of capital show that these marks served not

only to position leaves axes, but also control carving depth;

yet characteristic points on the side faces must also have

been dened. Asgari demonstrated how a threefold subdivi-

sion of the capital was used, step by step, to obtain its nal

form. In a Roman capital, the volutes could be produced

simply by trimming the corners to the required prole; in

a Hellenistic capital the same method was followed, but

with greater care to establish the exact positions of the key

features. Marking the half-height of either the capital or the

calathus could have been achieved arithmetically, while the

abacus furnished the principal vertical datum. The acanthus

leaves of both Hellenistic and Roman capitals were laid out

and carved in essentially the same way. Above the leaves,

however, the Hellenistic capital demanded far greater pre-

cision: unlike its Roman counterpart, its upper, exposed

zone gives full prominence to vegetal detail, leaving no

scope for error or for the carver’s whim. Any executional

deviations therefore had to be tiny. An appropriately pre-

pared template, used throughout the carving process, would

have allowed continuous three-dimensional checks.

If the capital’s design was an individual project, it must

nonetheless have obeyed certain rules – and those rules

probably did not rest on a mere arithmetic partition. Could

a geometric model have been employed? One therefore

asks whether the circular diameters that correspond to the

heights of the capital and the calathus might be linked to

still other subdivisions. A further issue is the possible in-

uence of Egyptian mathematical knowledge on the solu-

tions adopted by Ptolemaic scholars and architects (Rossi

2007). The literature on the geometric conception of pyra-

a b

The Corinthian capital of Hermopolis Magna (Egypt) and its geometric analysis

mids and temple plans is extensive; such expertise diered

from Greek practice, and we know that in Egypt Hellenistic

builders did not conne themselves to purely Greek-style

temples. The trac in mathematical ideas and architectural

concepts may thus have been two-way.

For more than a century scholars have examined ancient,

a well Gothic monuments and works of art in search of the

practical use of geometry – projecting designs and shaping

representations by means of geometric constructions, g-

ures, and their transformations. There are studies by Matila

Ghyka and Otto Hagenmaier as well Jay Hambidge, Ma-

cody Lund, Ernst Mössel set a special interest on applica-

tion of geometry and the “golden section”. Why not to ex-

amine a Capital from Hermopolis as a geometrical design?

The capital’s geometry

Attempting to test geometry of a capital, I had not consi -

dered Vitruvius description on a method establishing a size

of an abacus plinth as a real geometrical principle. In fact

his saying about a diagonal of an abacus square was a pure

geometry establishing its incommensurable length. It was

in contrary to an overall arithmetical system used for par-

tition of a capital. At that stage of the study, Vitruvius de-

scription had been considered as a simple and practical

meth od, without a deeper geometrical idea.

Geometry makes its appearance in the capital the mo-

ment a square is described upon a circle; conversely, a cyl-

inder can be inscribed in – and extracted from – the cube

whose side equals that square, as has been recognised for

Roman capitals. The system of successive squares, and

of the circles inscribed within them, could in principle be

extended ad innitum by applying the “duplication of the

square” construction discussed in Plato’s Meno: the diag-

onal of a square whose side is a (or 2a) is incommensu-

rable, measuring a√2 (or 2a√2). From the side of the ini-

tial square, therefore, one can generate a larger square and

inscribe within it a larger circle (Saito 1995; Rossi 2006,

88–90; Scott 2006). Other polygons can likewise be set

within the circle, binding them geometrically to the square

(Fowler 1999, 32–34).

In analysing the present capital, however, both circles

corresponding to the lower and the upper column diameters

– the capital height and the calathus height – had to be taken

into account, since their centres are displaced from one an-

other by one-half of the abacus module. The question then

arose: would that oset generate further relationships that

locate any characteristic points on the capital? It is not im-

possible that the capital’s dimensions were also partitioned

by means of the golden section, which Plato refers to as the

“mean ratio” (Ghyka 2001, 41–75; Naredi-Rainer 1982,

193–199; Hagenmaier 1977; Beutelspracher, Petri 1996;

Fowler 1999, 84, 85; Rossi 2007, 23–32, 67). This opens

up a considerable range of geometric possibilities – all

of which must be tested against the fabric of the capital itself.

The performed diagrams are accompanied by the fol-

lowing, two-level system of notation:

– General geometric rules are expressed with lower-case

symbols – for example a, b, 2a, 1/2a, a√3 – which refer

simply to lengths or operations in the abstract.

– Constructions specic to the capital are denoted by

capital letters and by colour:

○

A (red) designates every gure – side A, circle A,

red triangle, red square, red hexagon – drawn on the

lower diameter of the column shaft, that is, on the capi-

tal’s overall height. Diameter A consists of 2 radiuses a,

thus a is equal to 1/2A.

○

B (green) refers to the corresponding gures – cir-

cle B, side B, and so forth – constructed on the upper

diameter of the shaft, which equals the calathus height.

Diameter B consists of 2 radiuses b.

○

A second, equally important point: some geometric

operations are checked arithmetically, and the numerical

results are recorded in square brackets. This dual proce-

dure makes it possible to test whether the constructed di-

agrams are genuinely compatible with the mathematical

calculations based on the two fundamental dimensions

– the lower and upper shaft diameters, corresponding re-

spectively to the heights of the capital and the calathus.

In many instances the arithmetic values will not coincide

perfectly with the drawn gures, because each construc-

tion belongs strictly to magnitude A or magnitude B; the

gures may therefore dier by a minute margin. What

is striking, however, is the remarkable closeness of the

solutions that emerge.

When the geometric analysis began – an exploratory ex-

ercise at rst – the two circles, red and green, were traced on

the capital’s ground plan and squares were described upon

them. Their diagonals were then drawn, producing right-an-

gled triangles. That single subdivision already pointed to

a number of intriguing loci in the layout of the de

co ration

of the capital.

The next step was to construct two equilateral triangles,

with sides A and B, respectively, their bases laid along the

bottom edge of the capital (Fig. 3). Choosing the equilateral

triangle at the outset was a matter of chance; it is not that

the author was prompted by Plato’s dictum on the use of

that gure in generating the regular polyhedra that consti-

tute the fabric of the cosmos (Plato 2016, Timaeus 53c). At

this preliminary stage, moreover, I was unaware of Euclid’s

construction of an equilateral triangle inscribed in a circle.

The diagram was devised on the basis of nothing more than

elementary school geometry.

The very rst equilateral triangle, its side B placed on

the capital’s base, produced an apex that fell precisely on

the centre of the stalk of the central ower. This was an

interesting result – though one that might still have been no

more than fortuitous. The height marked by the rst equi-

lateral triangle proved noteworthy, for it coincides almost

exactly with the level of the centres of the corner volutes.

A second, analogous equilateral triangle with side A was

then drawn; its apex, only slightly higher, fell almost pre-

cisely on the underside of the abacus – an outcome no less

striking. [The height obtained for equilateral triangle B is

67.03 cm. The height of equilateral triangle A is 78.1 cm,

whereas the diameter of circle B equals 77.4 cm, a dier-

ence of merely 6–7 mm over the capital’s full height of

90.2 cm, with an average error of 0.66–0.77%].

The same procedure was applied to side C, correspond-

ing to the upper torus diameter of the base, and to side D,

Marek Barański

the lower torus diameter – eight and nine abacus units of

12.9 cm, respectively. The apex of triangle D clearly rose

above the capital, while that of triangle C lay very close to

the capital’s overall height. In the analyses that followed

the diameters associated with circles C and D were there-

fore set aside, and attention focused on the gures con-

structed for A and B.

For certainty, the apex levels of triangles A, B, and C were

recalculated arithmetically. Although they lie near one anoth-

er, dierences greater than a centimetre make it dicult to

accept them as points that coincide exactly with the capital’s

characteristic levels [apex of triangle C = 89.37 cm,

when

hight of the capital is 90–90.2 cm; apex of triangle D =

103.1–103.2 cm]. Even so, the close agreement between the

apices of triangles B and C and the heights of the top of

the calathus (77.4 cm) and of the whole capital (90.2 cm)

may well have oered the ancient designers a clue or guide-

line, despite the small error – about one per cent by modern

reckoning. We must bear in mind that ancient mathematics

could not compute irrational numbers with modern exac-

titude. Magnitudes were determined geometrically and

rounded with great, though not absolute, precision; milli-

metre-scale discrepancies would have been considered neg-

ligible. It is therefore plausible that, in Vitruvius’ simplied

description, such slight deviations were deemed unimport-

ant, allowing the outcome of this geometric construction

to be expressed as the simple, generalised partition of 1/7.

In my next operation, the equilateral triangles were

drawn in the reverse orientation, their apex now placed at

the centre of the capital’s base (Fig. 4). This time the api-

ces of equilateral triangle B marked not only the level of

the centres of the corner volutes [67.03 cm] but also, with

almost perfect accuracy, the reconstructed centres of the vo-

lutes themselves; the small error visible in the drawing is

probably the result of an inevitable imprecision in the recon-

struction. A further coincidence proved equally intriguing:

the side of triangle B intersected the smaller circle at a pair

of highly characteristic points – namely, those at which the

tangent line xes the inclination of the stems before it turns

into the corner volutes. We shall return to this matter later.

Surprised by these observations, I next drew a large

equilateral triangle with side A equal to the lower diameter

of the column shaft (Fig. 5). As might be expected, one side

of this triangle indicated almost exactly the lower face of

the abacus slab, a level that had already been xed by the

larger circle. What is more, it became evident that the larg-

er circle A, with its greater diameter, met – approximately

– the corners of the smaller triangle B, which had already,

again approximately, located the volute centres. This geo-

metrical diagram was at once unexpected and stimulating,

and it encouraged a search for additional constructions. At

this stage of the analysis some diagrams were produced

almost accidentally, yet the drawings themselves began to

suggest the next steps to be taken.

In the next stage of the investigation the construction of

right-angled triangles was tested (Fig. 6). As is well known,

two types of right triangle can be inscribed in a circle of

radius a being equal to 1/2A. The rst is isosceles, with

catheti measuring 1/2a√2 and a hypotenuse equal to the di-

ameter 2a; its apex coincides with the centre of the circle

and of the square that can be inscribed in it. The second has

Fig. 3. Construction of equilateral triangles of a side equal to A, B, C, D

diameters. Apexes of A and B triangles designates a characteristic

points and lines of the capital (elaborated by M. Barański,

drawing by K. Jezierski)

Il. 3. Konstrukcja trójkątów równoramiennych o bokach równych

średnicom kół A, B, C, D. Wierzchołki trójkątów A i B wyznaczają

charakterystyczne punkty i linie kapitela

(oprac. M. Barański, rys. K. Jezierski)

Fig. 4. Inversion of equilateral triangles A and B marks centers

of a corner volutes

(elaborated by M. Barański, drawing by K. Jezierski)

Il. 4. Odwrócenie trójkątów równobocznych A i B

wyznacza centra narożnych wolut

(oprac. M. Barański, rys. K. Jezierski)

The Corinthian capital of Hermopolis Magna (Egypt) and its geometric analysis

catheti a and 1/2a√3, with the same hypotenuse 2a. Both

varieties were drawn in circles A and B.

The apex of the larger, red triangle A fell almost exactly

on the centres of the volutes – points already indicated by

the height of the smaller, green equilateral triangle B. When

the shorter cathetus of this right triangle was set o along

its hypotenuse, which serves as the axis of our geometric

scheme, the distance obtained corresponded to the diameter

of the smaller green circle B. Thus yet another composite re-

lationship emerged, derived from the two diameters A and B.

At this moment memory supplied a schoolroom fact:

using right triangles – rather than merely stepping the com-

pass round the circumference – one can likewise construct

a hexagon inside a circle. Hexagons were therefore drawn

in both circle A and circle B. The outcome was no less

striking than before. The apex of the right triangle whose

hypotenuse equals diameter A and whose shorter cathetus

equals the radius a (1/2A) landed virtually in the centre of

a volute. As noted earlier, the volute-centres lie on the level

determined by the equilateral triangle of side B (67.03 cm);

consequently the longer cathetus of this right triangle ex-

actly equals side B. If the same right triangle is inverted, its

apex marks a distinctive point that is the lower vertex of the

hexagon inscribed in the red circle A. These lower vertices

establish the level of the capital’s lower acanthus leaves.

Fig. 5. Equilateral triangles inscribed into A and B circle.

Pairs of triangles A and B generate a hexagone A and B respectfully.

Apexes of a hexagon A marks characteristic points of a capital.

Triangles also designates a smaller inscribed circles

of 1/2A and 1/2B diameter, when a circle of 1/2A frames

a central decoration of the capital

(elaborated by M. Barański, drawing by K. Jezierski)

Il. 5. Równoboczne trójkąty wpisane w koła A i B. Pary trójkątów A i B

tworzą odpowiednio sześciokąty A i B. Wierzchołki sześciokątów

wyznaczają charakterystyczne punkty kapitelu. Trójkąty wyznaczają

również mniejsze wpisane koła o średnicach 1/2A i 1/2B,

gdzie koło 1/2A obejmuje centralną dekorację kapitela

(oprac. M. Barański, rys. K. Jezierski)

Fig. 6. Points designating by hexagon are mutual to points designated

by a right triangles of hypotenouse equal to a diameter of A and B

circles. Pairs of triangles construct rectangles A and B respectfully

(elaborated by M. Barański, drawing by K. Jezierski)

Il. 6. Punkty wyznaczone przez sześciokąt są również punktami

wyznaczającymi trójkąty prostokątne o przeciwprostokątnej równej

średnicom kół A i B. Pary trójkątów konstruują odpowiednio

prostokąty A i B (oprac. M. Barański, rys. K. Jezierski)

When the analogous right triangle B was constructed for

the smaller green circle, the results were less spectacular

but still signicant. The upper vertices of the smaller hexa-

gon B coincide, with high probability, with the terminations

of the middle blossoms and thereby with the junction of the

intermediate volutes; its lower vertices x the plane where

the lower leaves begin to bend outward.

The composite gure derived from superimposing the

two right triangles A and B not only provided the six vertices

of the hexagon: it produced a six-pointed star – the familiar

form generated by two equilateral triangles rotated relative

to one another and inscribed in the same circle. Inscribing

gures in a circle is, of course, no more than a consequence

of geometric law; yet this diagram yielded another, wholly

unexpected curiosity. The side of the equilateral triangle in-

scribed in circle A is exactly equal to the diameter of circle B.

No such coincidence had been anticipated, and it emphatical-

ly conrms the internal relationships between the two circles.

It was further observed that in this diagram an addition-

al conjunction of gures occurs: the red hexagon A almost

rests against the side of the green square B. This may be

no more than a chance juxtaposition, yet it could equally

be a consequence of the 7 : 6 ratio adopted for the two cir-

cles. An arithmetical check of the relevant dimensions re-

veals only minute discrepancies – something we must bear

in mind when analysing the geometry of such a scheme.

[The side of hexagon A is one-half of the 90.2 cm diameter,

that is, 45.1 cm. From this one can calculate the hexagon’s

“height” (the distance between two opposite sides), which

Marek Barański

is 78.1 cm. This is greater than the side of square B, which

measures 77.4 cm, the dierence being ca. 7 mm, thus an

average error is 1%. We should remember, however, that

the divergence between the side of a square and the side of

a hexagon constructed on the same centre amounts to only

half that value – i.e. 3.5 mm, thus an average error is signi-

cantly diminished].

The basic diagram was then expanded with a further set

of constructions. Attention was drawn to the paths taken by

the catheti of the two right-angled triangles. It turns out that

each cathetus is tangent to a subsidiary circle whose diam-

eter is 1/2A, and likewise to a second, still smaller circle

whose diameter is 1/2B. The centres of these two subsidi-

ary circles are displaced from one another by exactly half

an abacus unit of one-seventh of the capital’s height. The

diameters of the smaller circles coincide with the sides of

the hexagons inscribed in circles A and B. Even more tell-

ing is the fact that the 1/2A circle ts neatly into a rectangle

whose shorter side measures 1/2A and whose longer side

equals B; this longer side is the very cathetus that previous-

ly xed the volute centres and the level of the lower leaves.

The arrangement follows inevitably from the geometry of

the construction. A comparable conguration can be drawn

for the smaller 1/2B circle, although its enclosing rectan-

gle lacks such striking side-lengths. Remarkably, however,

the circle of diameter 1/2A denes the span of the capital’s

central decoration, and the 1/2B circle performs a similar

function – though within a slightly dierent zone.

Inscribing those right-angled triangles in the two circles

opened the way to further geometric operations. Two such

triangles can be combined to generate a rectangle whose

diagonal equals the diameter A being equal to 2 radiuses a,

whose shorter side measures a, and whose longer side mea-

sures 1/2a√3. If two of these rectangles are set perpendicu-

lar to one another, their overlap forms a square with side a;

within that square one may inscribe a circle of diameter a

equal to 1/2A. On the common axis of the larger circle

(of 2 radiuses a, being equal to A diameter) one can like-

wise erect two right-angled triangles whose vertices, when

joined, produce a hexagon with side a. That hexagon, how-

ever, can also be obtained in a dierent way: rst inscribe

an equilateral triangle in the circle, then add a second, in-

verted triangle of the same size. The resulting six-pointed

star is composed of twelve small equilateral triangles, each

having a side equal to one-third of the side of the origi-

nal triangle. The intersections of the star’s arms mark out

a smaller hexagon, within which yet another circle of diam-

eter a can be drawn.

All gures inscribed in the circle of diameter 2a or A,

thus share a common inner circle of diameter being equal

to radius a. What interests us here is that this apparently

“simple” geometry, carried out only with compass and

straight-edge, no longer yields simple arithmetic: the side

of the isosceles triangle inscribed in the circle is irratio-

nal (a√3), just as the hexagon described on the same circle

with a side of an identical length A√3. In the course of the

analysis we meet a constant passage from larger to smaller

gures and from rational to irrational magnitudes, only to

return again to rational ones – a chain of interlocking geo-

metric relationships. Although these observations produced

suggestive results and stimulating ideas, they did not yet

explain the capital’s overall composition.

Dening the subsidiary circles with diameters 1/2A and

1/2B – which embrace the central zone of the capital’s dec-

oration – drew attention to another possibility: constructing

a geometric progression based on the sequence a : a√2 : 2a

: 2a√2. The scheme relies on inscribing a square in a circle

and then inscribing a second circle about that square. Such

a relation was often exploited in the Middle Ages “dubbed

ad quadratum” (Naredi-Rainer 1982, 218, 219; Rossi 2007,

65, 66; Majewski 2013, 141, 142) and it reproduces the

“du plication of the square” that Plato sets out in the Meno

– NB this procedure is not connected to the famous proce-

dure of “squaring the circle”, which asks whether one can

construct, with only a straightedge and compass, a square

having the same area as a given circle (Hobson 1913, 14–

22; Rossi 2007, 67, 68). When this inverted diagram was

applied to the capital, the result was dizzying: almost every

corner of the progressively smaller squares seemed to coin-

cide with a signicant detail. Two steps, however, appeared

genuinely meaningful (Fig. 7). Beginning with circle A,

the rst operation yields square A

, inscribed in that circle.

The level of A

’s upper side corresponds, in broad terms,

to the lower edge of the abacus, though it is not identical

with it; that lower edge, as we know, is dened by the side

of square B described on the calathus circle B. The lower

vertices of A

rest on the tips of the capital’s lowest leaves.

A second iteration produces circle and square A

, each

with a diameter and side of 1/2A. Here the upper corners

of square A

support the lateral blossoms, and, as previous

constructions have shown, circle A

is identical with the

subsidiary circle obtained from the right-angled triangles

and from the equilateral triangle inscribed in the 2 A cir-

cle. Circle A

neatly encloses every element of the capital’s

central ornament. Repeating the same procedure for circle

B reveals that the vertices of square B

touch the inner edg-

es of the stems from which the corner volutes emerge. The

circle inscribed in B

likewise frames part of the capital’s

central decoration. A still smaller square, B

(side 1/2B),

has an upper side that marks the junction of the interme-

diate volutes, while its vertical sides intersect the stems at

mid-height. Whether these intersections truly determine the

inclination of the stems remains open to question; mere-

ly picking out conspicuous subdivisions and points on the

capital proved unsatisfying and did not yield a fully coher-

ent constructive schema.

To resolve this problem the following solution was pro-

vided (Fig. 8). Tangents were drawn along the outer edges

of the stems, and it became clear that – below, outside the

capital – these tangents meet at a single point. At the top they

also intersect the extensions of the sides of squares A and B.

Working rst with the B-dimension, I connected that lower

intersection‐point with the two points where the tangents

meet the prolongations of the sides of square B. The three

points form an isosceles triangle of striking height: it equals

two diameters of the smaller circle B, the two diameters

touching one another at the centre of the calathus – in other

words, the calathus height has been enlarged on each side

by the radius 1/2B. Discovering that the inclination of the

stems could be xed by a purely geometric construction

The Corinthian capital of Hermopolis Magna (Egypt) and its geometric analysis

was astonishing, but the surprise deepened when I repeated

the operation for circle A.

In the A-case the construction was inverted. Two cir-

cles of diameter A were drawn so that they touch at the ca -

pital’s centre. The lower intersection of circle A with the

ca pital’s vertical axis supplied the apex of an isosceles tri-

angle; from the upper intersection of the axis with the sec-

ond circle A I dropped a perpendicular to locate the base of

that triangle. When I joined the resulting base-points to the

extensions of the larger square’s sides, the full base-length

of the isosceles triangle A emerged. Connecting the apex

to those base-points left me speechless: the sides of the

larger and smaller triangles coincide with the thicknesses

of the stems visible in the drawing. One might have expect-

ed many things, but not so unambiguous a result. Even the

greatest sceptic must concede that the principles of geome-

try are intertwined with the design of the Corinthian capital;

this diagram is no accident but the product of a deliberate

construction.

The construction itself is not conned to the capital

block: it projects well beyond its outline and must have

been worked out geometrically rst, then reduced in scale

and transferred to the working drawing of the capital. The

inclination of the stem lines is determined by the sides of

an isosceles triangle whose height equals fourteen abacus

units and whose base measures seven such units. When that

inclination is carried onto the capital it appears as the hy-

potenuse of a right triangle whose legs are 7 and 1.75 aba-

cus units – or, expressed dierently, 7 units and 7⁄4 units.

Delving still further into the geometry, I prepared a se-

ries of additional diagrams, this time to test the possible

deployment of the irrational quantities √2 – the diagonal

of a square – and √3, which arises in the height of an equi-

lateral triangle, together with the next term √4 (Fig. 9).

The constructions were laid out on one-half of the sides of

squares A and B. Many of the points thus generated had al-

ready surfaced in the work with equilateral triangles, where

– as noted – there is a close correspondence between the

height of the A-triangle and the diameter/side of the B-cir-

cle-and-square.

A particularly instructive result emerged for the segment

1/2B. If the diagonal 1/2B√

is “folded down”, it xes the

level of the lower acanthus leaves – the very height previ-

ously determined by the vertices of the hexagon. The test

was repeated with the half-diameter of the smaller circle

– the distance that, as we have seen, governs the lower-leaf

height. The diagonal of the square on that half-diameter in-

tersects the capital’s axis at roughly the depth of the ini-

tial inward curl of the central blossom. By contrast, setting

o the longer irrational 1/2B√

gives the now-familiar

Fig. 7. Geometrical sequence of

a circle inscribed into a square

of A side. In a second step

a central circle measuring 1/2A

is reached, and this is

equal to radius a

(elaborated by M. Barański,

drawing by K. Jezierski)

Il. 7. Ciąg geometryczny kół

wpisanych w kwadraty o boku A.

W drugim kroku centralne koło

mierzące 1/2A zostaje

wyznaczone i jest ono

równe promieniowi a

(oprac. M. Barański,

rys. K. Jezierski)

Marek Barański

characteristic point: the apex of the small equilateral trian-

gle. Pushing one step further, 1/2B√

closes at the underside

of the abacus, completing the square constructed on the up-

per shaft diameter.

An analogous exercise was carried out for half the larger

diameter. The segment 1/2A√

marks the upper run of the

intermediate volutes and the seating of their outer stalks.

Extending to 1/2A√

yields a point very close to the di-

ameter of the subsidiary circle and almost coincident with

the height of equilateral triangle A (the calathus height is

77.4 cm, whereas the computed apex of the A-triangle is

78.1 cm – a divergence of about 0.7 cm, slightly larger than

one would wish, an average error is 0.9%). Plotting 1/2A√

revealed that the arc so described cuts the volute centres;

that same arc is also intersected by the line that constructs

1/2A√

, and it lies almost exactly on the volute centre de-

rived from √

of the smaller circle. Reversing the √

oper-

ation on the side B again located the lower-leaf level, but

performing the same reversal on the side A produced no

signicant result.

During the course of the investigation the golden-section

principle was also put to the test. Plato describes that pro-

portion, calling it aesthetically perfect (Plato 2016, Timaeus

31c–32a); Euclid sets out its construction in Book II, Prop-

osition 11, naming it “the extreme and mean ratio.” The

ratio was examined for the capital’s height by constructing

a right-angled triangle whose shorter cathetus measures

1/2A and whose longer cathetus equals A. The shorter side

proved to correspond, more or less, to the side of the abacus

slab as xed by Vitruvius’ rule. When the golden division

was marked on the hypotenuse, the point obtained coincid

ed with the level of the upper acanthus leaves – a level that,

in any case, results from halving circle B. A parallel test

on the right-angled triangle constructed for dimension B

yielded no signicant result.

This analysis therefore produced no conclusive evidence,

although one must remember that the golden ratio can arise

in other congurations, notably within the pentagon. Given

the paucity of ancient sources, it remains dicult to deter-

mine which additional constructions were actually familiar

in the Hellenistic period (Fowler 1999, 84, 85); many of

the more elaborate procedures were introduced much later,

some in the 20

century.

Here the diculty arises of deciding which geometric

constructions were actually known in the Hellenistic peri-

od and which were introduced only later, even in modern

times. Geometrically, the golden ratio can be demonstrated

in a right triangle whose sides have the requisite lengths;

by operating on a square and taking the diagonal of one of

the rectangles obtained by halving it; by forming a right

triangle whose vertices belong, respectively, to a pentagon,

a hexagon, and a decagon inscribed in circles of identical

diameter; by a construction based on the sides of a penta-

gram; and – most recently – by a procedure derived from

Fig. 9. Characteristic points of a capital are related to progression

of a side equal to radius (R) of circles A and B by a root of 2, 3, 4.

Minus “–” marks inverted progression

(elaborated by M. Barański, drawing by K. Jezierski)

Il. 9. Charakterystyczne punkty kapitela odnoszą się do postępu

kwadratu, sześcianu i czwartej potęgi budowanego dla promienia (R)

kół A i B. Minus „–” oznacza odwrócony postęp

(oprac. M. Barański, rys. K. Jezierski)

Fig. 8. Geometrical construction of two circles A and two circles B

designates both an angle of a corner stem leaning, as well as thickness

of a stem edge (elaborated by M. Barański, drawing by K. Jezierski)

Il. 8. Geometryczna konstrukcja dwóch kół A i dwóch kół B wyznacza

zarówno kąt pochylenia narożnych łodyg, jak i grubość krawędzi

łodygi (oprac. M. Barański, rys. K. Jezierski)

The Corinthian capital of Hermopolis Magna (Egypt) and its geometric analysis

an equilateral triangle inscribed in a circle, rst generated

geometrically in 1984 (Odom, van de Craats 1986).

The last of these, together with the pentagram-in-a-circle

diagram, is of particular interest in the present context: both

represent further examples of gures inscribed in a circle,

analogous to the hexagon and the square already considered.

A full discussion of the golden section and its possible con-

nection with the capital will demand a separate geometrical

analysis, but it is worth emphasising here the simple case of

an inscribed pentagon – a possibility not yet explored.

The pentagon – one of the forms employed in the con-

struction of Platonic bodies – was next tested. A pentagon

was inscribed in each of the two circles, A and B, that is,

on the lower and upper shaft diameters. When the gures

were set with their apices pointing upward they yielded

nothing of particular interest. It is worth noting, however,

that the arcs traced to generate the vertices of both penta-

gons A and B enter into special relationships that govern

the outlines and positions of key elements in the capital’s

central ornament, the zone delimited by the circle of di-

ameter 1/2A being equal to a radius a. A second diagram

was drawn with the pentagons inverted, their apices turned

downward (Fig. 10a). To my surprise, the base of penta-

gon B now established a horizontal line that links the stem

of the central blossom with the centres of the lateral blos-

soms; indeed, the centres of those side blossoms lie exact-

ly at the vertices of the same pentagon. The situation re-

calls an earlier result obtained with the equilateral triangle,

whose vertices located the centres of the corner volutes.

The pentagon thus proves to be a useful gure, even though

it identies relatively few characteristic points. An intrigu-

ing question now arises: might the very act of inscribing

a pentagon within a circle have been consciously exploited

in designing a Corinthian capital? Was the pentagon itself

a signicant gure in the original conception? One further

– and quite unexpected – feature of the geometry seems to

support such a view. We recall that inscribing a hexagon in

a circle produces a gure whose side equals half the cir-

cle’s diameter, that is, the radius on which the hexagon is

constructed. At rst sight one would assume that no com-

parable relation could apply to the pentagons employed in

the capital, for the side of a regular pentagon is by nature

larger than one-half the diameter of its circumscribed cir-

cle. Yet a surprising equivalence emerges when the side

of pentagon B is compared with one-half the diameter of

circle A: the two measures coincide precisely (Fig. 10b).

Such concordance can hardly be accidental; rather, it is the

result of a perfect calibration between the dimensions of the

two circles. We have here yet another instance of the way

the entire network of gures – essentially generated from

a single segment – locks together, taking full account of the

special reciprocal relations between the circles that corre-

spond to the lower and upper diameters of the column shaft.

Let us now return to our basic geometric operations and

consider them in terms of rationality, irrationality, and the in-

ter-relations of the gures. The rst gure was an equilateral

Fig.10. Geometric analysis of capital: a) pentagram inscribed into A and B circle. An upper corners of B pentagon designates centers of side

flowers, b) side of hexagon inscribed into A circle is equal to a side of pentagon inscribed into B circle. Thus we can construct a capital scheme

being on a line segment equal to 1/2 of A circle or radius a. Circles A and B are in proportion 7 to 6, thus a diferences is 1/7A and this is equal

to abacus height (elaborated by M. Barański, drawing by K. Jezierski)

Il. 10. Geometryczna analiza kapitela: a) pięciokąt wpisany w koła A i B. Górne naroża pięciokąta B wyznaczają środki bocznych kwadratów,

b) bok sześciokąta wpisanego w koło A jest równy bokowi pięciokąta wpisanego w koło B. W ten sposób możemy konstruować schemat kapitela

będącego opartym o odcinek równy połowie średnicy koła A, będącym promienieniem a. Koła A i B mają proporcje 7 i 6, co daje różnicę 1/7A,

a to jest równe wysokości abakusa (oprac. M. Barański, rys. K. Jezierski)

a b

Marek Barański

triangle resting on the capital’s base, its side equal to the

diameter of circle A. That produced a second equilateral

triangle inscribed in the same circle; its side is irrational,

yet within that triangle one can inscribe a circle whose di-

ameter is 1/2A. A striking conjunction emerges: the verti-

cal distance between the apex of the large triangle (side A)

and the apex of the inscribed equilateral triangle is, to

a close approximation, 1/7 of the diameter of circle A – in

other words, the abacus height. Both 1/7A and 6/7A are ir-

rational magnitudes, but, as we have seen, 6/7A is the di-

ameter of circle B. By further operations on circle B we

can generate new gures that interlock successively within

circles A and B.

Superposing two equilateral triangles in circle A produc-

es a six-pointed star whose outer vertices dene a hexagon

with side 1/2A; that hexagon can in turn accommodate, al-

most exactly, a square whose side is B. It is remarkable that,

starting from a triangle with an irrational side, we obtain

a hexagon whose side is the rational segment 1/2A – and

that both triangles and the hexagon can each contain a cir-

cle of diameter 1/2A.

Right-angled triangles inscribed in circle A, with the

diameter as hypotenuse, yield two further constructions

depending on the catheti chosen. In the rst case the right

triangle is half of the square inscribed in the circle; as per

Plato’s Meno, this square has an irrational side. Within it

one can inscribe a circle of diameter 1/2A, which recurs

in the geometric progression founded on alternately in-

scribing circles and squares: in a circle of diameter 2a we

inscribe a square of side a√2; within that square a second

circle of the same diameter; and then a new smaller square

and circle of diameter a.

A six-pointed star (and hence a hexagon) can also be

obtained from the vertices of right-angled triangles whose

hypotenuse equals A, whose shorter cathetus is 1/2A, and

whose longer cathetus is A√3. Combining two such trian-

gles gives rectangles with one irrational side and one side

1/2A, each rectangle admitting a circle of diameter 1/2A;

three such rectangles together outline a hexagon of side

1/2A. A pentagon with irrational sides relative to circle

A can likewise be inscribed in that circle, and further g-

ures can be placed in circle B; each generates smaller but

ana logous forms and, with them, new relationships. The

Her mopolis capital is therefore saturated with just such

nested, mutually dependent congurations.

Conclusions

By working with two such schemata – the larger and

the smaller, corresponding respectively to the lower and the

upper diameter of the column shaft – the designer gained

a range of options that could individualise any given cap-

ital. A further point must be stressed when we examine

the geometry deployed in the diagrams discussed above:

the gures employed mesh with one another, forming in-

ter-locking constructions that seem to be the product of

a deliberate manipulation of geometric relationships. The

two systems based on the circles are displaced from one

another by exactly half a unit of the abacus height; in oth-

er capitals additional geometric constraints may well have

dictated a dierent oset. The model adopted here embrac-

es gures generated from circles, squares, triangles, penta-

gons, and hexagons. Constructed on the lengths A and B,

these gures mark the characteristic points of the capital’s

ornament, and the model readily incorporates the irrational

magnitudes that arise from them. Only after considerable

time did I grasp what Vitruvius really meant when he de-

scribed the method for determining the size of the abacus

slab. At rst I failed to grasp the import of the seemingly

simple instruction that “two diameters of the circles form

the diagonal of the square abacus”: it appeared to lack any

concrete geometric meaning. Once I began to work routine-

ly with geometric models – including recursive progres-

sions – it became clear that Vitruvius’ words are a practical

shorthand for an entire sequence of geometric proportions

generated from a circle’s diameter and the square erected

upon it. Measuring the size of the abacus by setting o a di-

agonal equal to two circle diameters signals that a circle

can be inscribed within that square, and that within that cir-

cle one may in turn draw the square whose side equals the

primary diameter. The same geometric logic underlies the

scheme that xes the inclination of the stems: there, too, we

have an isosceles triangle whose height equals two diame-

ters. Signicantly, the basis for constructing that isosceles

triangle is identical in form and dimension to the proce-

dure used to establish the abacus size; the only dierence

lies in the location – on the capital’s axis rather than on the

diagonal. Nothing here is accidental or idiosyncratic: each

element is a geometrically intelligible step in a single pro-

portional sequence.

The inter-relations between circles A and B do not, by

themselves, explain how the actual diameters were chosen

– diameters that represent the lower and upper widths of the

column shaft, or, in other words, the total height of the capi-

tal subdivided into abacus and calathus. The diameter of the

smaller circle B can, of course, be derived from the larger

circle A by a straightforward application of Thales’ theo-

rem; yet other methods are available. As we have already

shown for the Hermopolis capital, the apex of an equilateral

triangle whose side equals A – or, equivalently, the apex ob-

tained by setting out √3 on the half-side 1/2A – falls almost

exactly on the level of the abacus slab and thus determines

the diameter of circle B. That construction yields a division

that comes very close to the canonical ratio of one-seventh.

There is, however, an even more elegant way to obtain

both circles at once in the desired 6 : 7 ratio. In this solution

neither circle is derived from the other; instead, both are

constructed simultaneously from a single chosen segment

line (Fig. 10b). The key observation is that the side of the

hexagon inscribed in circle A is equal to the side of the pen-

tagon inscribed in circle B. On that basis one can devise

a geometric construction that xes the diameters of both

circles by starting with a segment equal to the order-mod-

ule – that is, the radius of circle A. The dierence between

the two diameters then corresponds exactly to one-seventh

of the diameter of circle A. Recognising this dependence

suggests that a dual system may have been developed, anal-

ogous to the method of locating the vertices of a hexagon by

halving its side, or to the constructions that employ right-an-

gled or equilateral triangles erected on half the diameter

The Corinthian capital of Hermopolis Magna (Egypt) and its geometric analysis

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–

namely, the radius – of the fundamental circle. All regu-

lar geometric gures can be constructed from a single seg-

ment or line (Majewski 2013, 108–112, 116). But our study

pointed out relations among gures inscribed into circles,

being constructed this way, on a basic segment, a module

responsible for a general architectural design, as well as

governing decoration of a crucial element – a capital.

The geometric model here reconstructed for the Corin-

thian capital from Hermopolis Magna was subsequent-

ly applied to other, earlier Hellenistic capitals of a simi-

lar form

– among them those of the Tholos at Epidauros

(ca. 360 BC), the Asklepieion in Athens (ca. 350 BC), and

the Propylon on Samothrace (ca. 260 BC). The preliminary

tests conrmed that the same underlying principles were in

force, even though the individual capitals reveal a dier-

ent choice of geometric gures – a variation that appears

to reect their diering proportions The earliest examples

exploit the interlocking relations of the gures in quite

a straightforward manner, whereas the later ones display

the system in a more elaborated form.

The issue of geometrical concept applied to Corinthian

capital design has to be discussed in a wider context. At that

moment we can only mention it perfectly ts to Phytagorean

philosophy, as well as achievements of ancient geometry of

this age. The studies of Theatetus of Athens (417–369 BC)

explains concept of a chosen basal segment line (Król 2006,

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ciple being used in a Corinthian capital design of Hellenistic

period Author is preparing currently a study “Early Corin-

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orean perfection” explaining development of the concept.

Let us return once more to the capital’s proportions

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cisely the dierence between the lower and the upper diam-

eters of the column shaft – exactly as Vitruvius describes

1:7. On the other hand, the lower diameter is the module of

the order, the basis of every proportional scheme in a Greek

temple. Thus it would be possible to dene proportion

scheme mentioning only partition of a main circle, but not

of two units. Referring to two dierent units inevitably in-

vites more error than working with a single measure. It is

striking that our ancient author did not put the matter more

simply – explaining a partition calculation as being based

on a module only. In the source elaborated by Hellenistic

architect, on which Vitruvius presumably relied, distingui-

shing the two diameters must have been crucial.

Translated by

Patryk Jan Bartuś

Marek Barański

Streszczenie

Koryncki kapitel z Hermopolis Magna (Egipt) i jego geometryczna analiza

W artykule zaprezentowano studium hellenistycznego korynckiego kapitela (240 p.n.e.) z Hermopolis Magna w Egipcie. Witruwiańskie propor-

cje zostały potwierdzone tylko częściowo. Spowodowało to podjęcie analizy geometrycznej. Wykonanie geometrycznych diagramów potwierdziło,

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Marek Barański

doi:10.37190/arc260301

Streszczenie