Michele Sbacchi
University of Palermo Dipartimento Storia e Progetto nell'Architettura Palazzo Larderia, Corso Vittorio Emanuele 188 90133 Palermo ITALY
Stoicheia
(Elements). This overwhelmingly influential text deals
with planar geometry and contains the basic definitions of the
geometric elements such as the very famous ones of point, line
and surface: "A point is that which has no part;" "Line
is breathless length;" "A surface is that which has
length and breadth only" [Euclid 1956, I:153]. It also contains
a whole range of propositions where the features of increasingly
complex geometric figures are defined. Furthermore, Euclid provides
procedures to generate planar shapes and solids and, generally
speaking, to solve geometrical problems. Familiarity with the
Elements allows virtually anyone to master the majority
of geometrical topics. Although all this is well known I nevertheless
find it necessary, given our misleading post-Euclidean standpoint,
to underline that 'Euclidean Geometry' was 'Geometry' tout
court until the 17th century. For it wasonly from the second
half of 17th century that other branches of geometry were developed—notably
analytic and projective geometry and, much later, topology. Yet
these disciplines, rather than challenging the validity of Euclidean
geometry, opened up complementary understandings; therefore they
flanked Euclid's doctrine, thus confirming its effectiveness.
In fact, Euclidean geometry is still an essential part of the
curriculum in high schools worldwide, as it was in the quadrivium
during the Middle Ages. That is not to say that Euclid's teaching
has never been questioned. In fact a long-standing tradition
does not necessarily imply a positive reverence: some Euclidean
topics have, indeed, undergone violent attacks and have fostered
huge debates. The ever-rising polemic about the postulate of
the parallels is just one notorious example of the many controversies
scattered throughout its somewhat disquieted existence.Euclid was far from being an original writer. Although conventionally
referred to as the inventor of the discipline, he was hardly
an isolated genius. Historians of mathematics have clarified
how he drew from other sources—mainly Theaetetus and Eudoxus.[1] Hence,
rather than inventing, he mostly systematized a corpus of knowledge
that circulated among Greek scholars in somewhat rough forms.
Therefore, Euclid's great merit lies in the exceptional ability
to illustrate and synthesize.. Although marred by contradictions
and gaps, the
Elements. It is nevertheless true that the other
branches of geometry, which arose from the 17th century on, affected
architecture, but this can be considered a comparatively minor
phenomenon. In fact, the influence exerted by projective geometry
or by topology on architecture is by no means comparable to the
overwhelming use of Euclidean geometry within architectural design
throughout history.The relevance of Euclidean methods for the making of architecture has been recently underlined by scholars, especially as against the predominance of the Vitruvian theory. According to these studies [Rykwert 1985; Shelby 1977], among masons and carpenters Euclidean procedures and, indeed, sleights of hand were quite widespread. Although this building culture went through an oral transmission, documents do exist from which it can be understood that it was surely a conscious knowledge. 'Clerke Euclide' is explicitly referred to in the few remaining manuscripts.[3] Probably the phenomenon was much wider than what has been thought so far, for the lack of traces has considerably belittled it. We can believe that during the Middle Ages, to make architecture, the Euclidean lines, easily drawn and visualized, were most often a good alternative to more complicated numerological calculations. Hence we can assume that an 'Euclidean culture associated with architecture,' existed for a long time and that it was probably the preeminent one among the masses and the workers. Yet among the refined circles of patrons and architects the
rather different Vitruvian tradition was also in effect at the
same time [Rykwert 1985:26]. This tradition was based on the
Pythagorean-Platonic idea that proportions and numerical ratios
regulated the harmony of the world. The memorandum of Francesco
Giorgi for the church of S. Francesco della Vigna in Venice,
is probably the most eloquent example illustrating how substantial
this idea was considered to be for architecture [Moschini 1815,
I:55-56; Wittkower 1949:136ff]. This document reflects Giorgi's
Neoplatonic theories, developed broadly in his If we now compare again these procedures with the Euclidean
ones, it appears more clearly that the difference between the
two systems is a significant one: according to the Vitruvian,
multiplications and subdivisions of numbers regulated architectural
shapes and dimensions; adopting Euclidean constructions, instead,
architecture and its elements were made out of lines, by means
of compass and straightedge. The 'Pythagorean theory of numbers'
and the 'Euclidean geometry of lines' established thus a polarity
within the theory of architecture.[5] Both disciplines were backed up and,
in a way, symbolized by two great texts of antiquity: the I have outlined how, during the Middle Ages, Euclidean and
Vitruvian procedures empirically coexisted within building practice.
This situation would undergo an important change in the 17th
century. During the Renaissance the advent of an established
written architectural theory, based as it was on the dialogue
with Vitruvius' text, fostered the neo-Pythagorean numerological
aspect of architecture. Leon Battista Alberti, the most important
Renaissance architectural theorist, was well aware of Euclidean
geometry,[7]
a discipline which he dealt with in one of his minor works, the
Ludi Mathematici. Yet Alberti's orthodox position within the
Classical tradition could not allow him to challenge the primacy
of numerical ratios for the making of architecture. Therefore,
not surprisingly, Euclidean methods are left out of his
Architettura
Civile, written presumably between 1670 and his death, marks
a fundamental moment of the relationship between Euclidism and
theory of architecture. But first, a reflection on Guarini's
activity allows us to understand that his being the first to
include Euclidean geometry extensively within an architectural
treatise was no accident. I do not want to dwell upon his general
involvement with geometry and the vast use of geometrical schemes
for his buildings, two issues doubtlessly but loosely related
to this fact. I would rather point out more circumstantial events.
Firstly, being a professor of mathematics, Guarini was almost
unavoidably obliged to consider Euclidean geometry. His Euclidean
interests probably arose during his early teaching of Mathematics
at Messina where distinguished Euclidean scholars such as Francesco
Maurolico and his pupil Giovanni Alfonso Borelli had taught previously.
There Guarini found himself in one of the most stimulating scientific
centers of the time where a long-standing Euclidean tradition
existed.[8]
Maurolico wrote a commentary of the Elements, [9] while Borelli was author of the Euclides
Restituitus. Yet it was more likely in Paris, where Guarini
taught mathematics between 1662 and 1666, that his concern with
Euclidean geometry expanded. For there he encountered a lively
scientific milieu and particularly Francois Millet de Chales.
A most distinguished mathematician, this latter was the author
of Cursus seu mundu mathematicus, an encyclopedic work
on mathematics that also dealt with architecture.[10] More relevant to the present discussion
are Millet's two commentaries on the Elements, Les
Huit Livres d'Euclide and Les eléments d'Euclide expliqués
d'une maniere nouvelle et trés facile. Guarini was
deeply influenced by Millet [Guarini 1968:5, note 1]; he is referred
to frequently in Guarini's books, not just for geometrical or
mathematical matters. Out of this background developed Guarini's
magnum opus on geometry, the Euclides Adauctus et methodicus
mathematicaque universalis published in 1671. As the title
makes clear it, was both a commentary on the Elements
and an attempt to summarize the mathematical knowledge of the
time, much in the manner of his beloved Millet. It turned out
to be a rather successful book for it was republished five years
later. Guarini, therefore, falls well within the tradition of
Euclidean commentators. His interest for the discipline went
beyond the mere content, however, as Euclidean geometry was for
him a sort of universal key for human knowledge. The extent to
which Guarini considered Euclidean norms as the basis of every
scientific work is also clear from another work of his, the Trattato
di Fortificazione, where the Euclidean basic definitions
of point, line, etc. are provided at the very beginning as a
kind of conditional entry to the topic.[11] The same approach occurs with his Del
modo di Misurare le fabbriche, a booklet on surveying.
Architettura Civile, sets forth a
system for the orders that is, even more geometrical than Guarini's.
Of course Osio's ideas, probably regarded as unorthodox or extravagant
by others, strongly appealed Guarini.[15] Hence, it is hardly surprising that
Osio, despite being a rather obscure architect, is taken by Guarini
as a primary authority, second only to Vitruvius, and is continuously
quoted throughout his Architettura Civile. With Guarini and Osio,
therefore, the Euclidean heritage is consciously acknowledged
within the learned realm of theory and no longer belongs to an
oral and empirical culture. Osio's Euclidean opposition to numerology
is clearly self-confessed: in the preface of his book he describes
the difficulties of the traditional modular systems: ".......such
those that (perhaps in order to avoid subdivisions that are intricate
in themselves) follow the fashion of the more modern with the
establishment of the modules, in which, relying on the discreet
property of the numbers.....".[16] And he then states that his method
will avoid the modules used by architects before him: "Thus
henceforth it always appeared that these were the possible ways,
and the only ones capable of putting in proportion the quantities
of the same order, both in themselves and amongst themselves.
And still in any case, through divine favour, I hope in this
work of mine to enrich Architecture to more certain and more
perfect effect. With Geometrical rules, which have for their
basis and support the Euclideian Demonstrations, I hope to aid...".[17] His new
attitude is also emphasized by a symbolic representation: in
the frontispiece he is significantly portrayed with two books
bearing the names of Vitruvius and Euclid, alluding unambiguously
to the double tradition I have outlined so far. Just as conscious
and deliberate is Guarini's Euclidism. Indeed Architettura
Civile turns out to be a rather peculiar trattato where Euclid
and Millet de Chales—two geometers—are advocated as
architectural authorities, even in the most quintessentially
architectural parts.[18]
The Euclideian leaning is revealed by a number of other circumstances.
In Architettura Civile quite often the elements of geometry
become the elements of architecture tout court. For Guarini,
for example, a wall is a 'surface' and a dome a 'semisphere.'
Consequently, 'architectural design' most often seems to be identified
with 'architectural drawing': as a true geometer Guarini describes
the production of the project rather than the production of the
building. In contrast to the two treatises of his pupil Vittone,
where technical problems are preeminent, Guarini's Architettura
Civile completely disregards the constructional aspect of
architecture in favor of detailed descriptions of drawing techniques.
This is striking, especially if we think of the technological
emphasis often displayed in Guarini's buildings. In this regard
it is curious that drawing tools are in fact grouped under the
title "Architectural Instruments". The problem, for
him, was not 'how to build' but 'how to draw.' Therefore, not
only Euclidean geometry has become a part of architectural theory
but it has also carried with it its implied linearis essentia
(linear-like essence) which in Guarini and Osio pervades the
all matter.The expression linearis essentia is Francesco Barozzi's. An
outstanding mathematician and friend of Daniele Barbaro, Barozzi
was the leader of a movement of general reappraisal of Euclidean
geometry, which centered around Barozzi in Venice and Padua and
around Federico Commandino in Urbino.[19] The achievements of this group of scholars
are essential to understanding how Euclidean geometry passed
from Serlio's timid acknowledgement to Guarini's broad inclusion
within architecture.[20]
Barozzi, Barbaro, Commandino and their circles contributed to
the recognition of geometry as a modern science. Consequently
they took the rigorous rereading of the Euclidean text as a conditional
starting point. Commandino dedicated all his life to retranslating
and clarifying Greek texts on science, among them the
To this Kepler replied in the
Judith Field has pointed out that "... the weight of
the geometrical work in
In geometry the approach is totally different: the entities adopted—line, volume, etc.—are thought of as continuous; they match the continuity of reality in a more comprehensive way than the discrete ones do. For example the geometrical line—not coincidentally taken as the symbol of the "continuous"—represents mensurable as well as incommensurable quantities, by means of the infinite series of his points. As a matter of fact the argument about discrete and continuous quantity has historically often been used to distinguish geometry from arithmetic, and sometimes to support the superiority of one over the other.[23] Geometry, in fact, often became synonymous with continuous. Mathematicians such as Barozzi, Tartaglia or Viviani—just to quote those from the period with which I have mainly dealt—were well aware of this distinction, as scientists are today. Architects, instead, only vaguely considered it. The very learned Scamozzi and the rather minor figure Osio are two of the few who included this topic, although very briefly, in their treatises. Guarini, who as a mathematician and philosopher discusses at length quantitas, continua and quantitas discreta in his books, disregards it almost completely in his architectural treatise.[24] This is rather surprising because, as I have tried to demonstrate, the field of architecture was a crucial battleground for the two conceptions. Indeed in the making of architectural forms the choice between a line to trace—i.e. the geometical approach—or a number to calculate—i.e. the numerological approach—not only implies rather different design methods but also brings about diverse results. The opposition of the continuous to the discrete enlightens how deep, conceptually, was the opposition of geometry to arithmetic. The change that occurred in architecture at the end of the 17th century, which witnessed a dismissal of Pythagorean numerology in favour of a more explicit adherence to geometry, is therefore a meaningful phenomenon. It consisted in making official rather widespread but disguised procedures. Furthermore, its belonging to a vast cultural phenomenon—of which I have analyzed the revival of Euclidean geometry within Italian scientific circles and Kepler's approach in the fields of astronomy and music—further magnifies its importance.
[2] Heath has pointed out that a Latin translation,
earlier than Adelard's, must have been the common source for
at least three documents: Boethius, a passage in the [3] Two manuscripts are located in the King's Library
of the British Museum, the Regius manuscript and the Coke manuscript.
See [Knoop 1938; Euclid 1956, I: 95; Halliwell: [4] "Proportio est ratae partis membrorum in omni
opere totiusque commodulatio, ex qua ratio efficitur symmetriarum,"
[Vitruvius, III, 1, 1]. [5] Girolamo Cardano stigmatizes this opposition when
in his [6] Mario Vegetti has written, "The tradition of
the [7] Alberti owned a copy of the [8] Note XVII of Michel Chasles' [9] Unpublished manuscript at the Biblioteque Nationale,
Paris. He also translated Euclid's [10] On Millet de Chales and 17th century encyclopedism
see [Vasoli 1978]. [11] "The Elements of Euclid are so necessary to
every science…and also to whoever would advance themselves
in the military arts must believe them to be the basis, principle
and fundamental element on which to build, and beyond which to
advance, and on which to lay every speculation" ("Gli
Elementi di Euclide sono si necessari ad ogni scienza ... e pertanto
qualunque vuole avanzarsi nell'arte militare, deve credere, che
questa sia la base, il principio & il primo elemento, di
cui si compone, e sopra a cui s'avanza, e cresce ogni sua speculazione")
[Guarini 1968: 10]. [12] "E perché l'Architettura, come facoltá
che in ogni sua operazione adopera le misure, dipende dalla Geometria,
e vuol sapere almeno i primi suoi elementi, quindi é che
ne' seguenti capitoli porremo que' principi di Geometria che
sono piú necessari" [Guarini 1968:10]. It is noteworthy
that Guarini defines geometry as [13] There were, in fact, two tradition for Geodesy.
The first referred to the lost treatise by Euclid on The Division
of Figures, of which existed an Arabic copy by Muhammed ibn Muhammed
al Bagdadi, translated into Italian in 1570. The second referred
to the Metrics of Hero. See [Guarini 1968: 389, n. 1]. [14] "The diffidence of pure geometry with regards
to logarithms" ("la diffidenza del puro geometra nei
confronti dei logaritmi.") [Guarini 1968: 418, n. 4]. [15] The acquaintance between Guarini and Osio is a likely
one. Guarini often visited Milan, Osio's town, to meet the publisher
of his astronomical work [16] "...come quelli pure li quali (forse per isfuggire
le sudette per se stesse intricate subdivisioni) doppo i piú
moderni con lo stabilimento dei moduli, ne quali appoggiantesi
alla discreta proprietá dei numeri" [Osio 1661: 2].
[17] "Laonde parve sempre da qui a dietro che questi
fossero i modi possibili, e unici di proporzionare le quantitá
nei medesimi ordini, tanto in se stesse quanto tra loro. E pure
ad ogni modo, mediante il favore divino, io spero in questa mia
opera, arricchire l'Architettura a questo effetto piú
certa e piú perfetta. Con regole Geometriche, ch'hanno
per loro base, e sostegno le Dimostrazioni Euclideiane, spero
agevolare....." [Osio 1661:2]. [18] See [Guarini I,1] where Millet is strikingly quoted
together with Vitruvius for the definition of architecture; and
I, III, Osservazione 6, where Millet is quoted for the matter
of the respect of ancients' rules; see also III, 17, 2, where
the topic is the Doric order. [19] Daniele Barbaro is quoted together with Vettor Fausto
and Nicoló Tartaglia as a restorer of the antique scientific
rigor in the dedication of Guidobaldo del Monte, [20] To this might be added John Dee's inclusion of architecture
among the mathematical arts. [21] The connection between syllogism and geometrical
reasoning was known since Socrates' times. See [Mueller: 292ff].
[22] A good summary is given by [Evans 1957]. See also
[Manin 1982]. [23] A position like that of Ramus is to this respect
symptomatic. On Ramus and French anti-Euclidism see [Bruyere
1984]. [24] Guarini gives this topic primary importance. His
Euclides begins with
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