University of Palermo
Dipartimento Storia e Progetto nell'Architettura
Palazzo Larderia, Corso Vittorio Emanuele 188
90133 Palermo ITALY
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 sourcesmainly Theaetetus and Eudoxus. 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, in its time, represented a gigantic step forward, especially compared to the fragmentary way in which geometry was known and transmitted. It soon became an immensely useful text for all the fields where geometry was applied. Optics, mensuration, surveying, navigation, astronomy, agriculture and architecture all benefitted in various ways from a newly comprehensive set of rules able to overcome geometrical problems. As its popularity grew, the Elements went through several translations. Following the destiny of most Greek scientific texts, it was soon translated into Arabic and was known through this language for almost fifteen centuries. A well-known Latin translation was made by Adelard of Bath in the 12th century but at least another translation existed earlier. Campano's Latin translation of 1482 was the first to be published. Nevertheless a translation directly from Greek into Latin was made by Bartolomeo Zamberti in 1505. Federico Commandino's Latin edition of 1572 was to become the standard one. The first English translation is due to Henry Billingsley in 1570, with a preface by John Dee [Wittkower 1974:98; Rykwert 1980:123]. No less significant are the commentaries upon the Stoicheia, if only because they witness the continuous debates that scholars engaged in about the text. Certainly the most renowned commentary is the one made in the 5th century A.D. by Proclus on the First Book. Because of this vast and lasting tradition, the Elements may be appropriately compared to the Bible or to the Timaeus as a cornerstone of Western culture [Field 1984:291].
ARCHITECTURE THEORY, GEOMETRY AND NUMBER
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. 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 De Harmonia mundi totius, published in Venice in 1525, which, together with Marsilio Ficino's work, can be taken as a milestone of Neoplatonic cabalistic mysticism. The whole theory, whose realm is of course much wider than the mere architectural application, was built around the notion of proportion, as Plato understood it in the Timaeus. Furthermore, it was grounded on the analogy between musical and visual ratios, established by Pythagoras: he maintained that numerical ratios existed between pitches of sounds, obtained with certain strings, and the lengths of these strings. Hence, the belief that an underlying harmony of numbers was acting in both music and architecture, the domain respectively of the noble senses of hearing and of sight. In architecture numbers operated for two different purposes: the determination of overall proportions in buildings and the modular construction of architectural orders. The first regarded the reciprocal dimensions of height, width and length in rooms as well as in the building as a whole. The second was what Vitruvius called commodulatio. According to this procedure, a module was established generally half the diameter of the column from which all the dimensions of the orders could be derived. The order determined the numerical system to adopt and, thus, every element of the architectural order was determined by a ratio related to the module. Indeed it was possible to express architecture by an algorithm [Hersey 1976:24]. Simply by mentioning the style a numerical formula was implied and the dimensions of the order could be constructed. These two design procedures are both clearly governed by numerical ratios series of numbers whose reciprocal relationships embodied the rules of universal harmony.
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. Both disciplines were backed up and, in a way, symbolized by two great texts of antiquity: the Timaeus and the Elements. Although in architecture the dichotomy was brought about substantially by the issue of proportion, the difference is, in fact, a more general one. Every shape and not only proportional elements can be determined either by the tracing of a line or by a numerical calculation. This twofold design option is somehow implied in the epistemological difference between geometry and arithmetic. Socrates' remark, in Plato's Meno, to his slave who hesitated to calculate the diagonal of the square, epitomizes the two alternatives: "If you do not want to work out a number for it, trace it" [Plato Meno 84].
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, 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 De Re Aedificatoria, where he quite decidedly states that: " ... the three principal components of that whole theory [of beauty] into which we inquire are number (numerus), what we might call outline (finitio) and position (collocatio)" [Alberti 1485:164v-165]. For him numbers were still the basic source. Accordingly, his seventh and eighth books, fundamental ones of De Re Aedificatoria, are devoted to numerical topics. Yet it might be speculated that his emphasis on lineamenta (lineaments) and lines, never fully understood, could be an acknowledgement of a building practice leaning more toward geometry than toward numerology. With Francesco di Giorgio Martini's Trattato di Architettura Civile e Militare, the Euclidean definitions of line, point and parallels make their first appearance within an architectural treatise, although in a rather unsystematic way. Serlio, later, goes a step further: his first two books include the standard Euclidean definitions and constructions; yet they are intended to be the grounds more for Perspective than for Architecture. Traces of Euclidean studies can be found also in Leonardo: the M and I nanuscripts, the Foster, Madrid II and Atlantic codices contain Euclidean constructions and even the literal transcription of the first page of the Elements [Lorber 1985:114; Veltman 1986].
Architettura Civile came later; it was definitely written after the Euclides since the latter is mentioned in it. As I have suggested, the Euclidean intrusions in Architettura Civile are far too many to justify them only on the grounds of a mere unconscious professional bias. The argument that the geometer prevailed over the architect misses the importance of the issue. In the first treatise of the five constituting the book, Guarini early on states his geometrical interests: "And since Architecture, as a discipline that uses measures in every one of its operations, depends on Geometry, and at least wants to know its primary elements, therefore in the following chapters we will set out those geometrical principles that are most necessary". Consequently the following chapter explores the "Principles of Geometry necessary to Architecture." It contains the nine definitions of point, line, surface, angle, right angle, acute angle and parallel lines. Chapters dedicated to surfaces, rectilinear shapes, circular shapes follow and the whole first treatise continues basically in this way with postulates, other principles and several typical Euclidean transformations such as "To draw a line from a given point in order to make it touch the circle" [Guarini 1968:41]. The Euclidean discipline of Geodesia fills the Fifth Treatise the way of dividing and transforming planar shapes into other equivalents. Some of these parts are literally transported from his own Euclides, some are slightly elaborated on in light of their architectural application. Guarini's Euclidean purismas opposed to arithmeticsis remarkably evidenced, when, in the Geodesia treatise, he considers progressions as purely geometrical and not numerical [Capo 8]. The dismissal of numerical progression, an attitude taken also by Francois Derand, was shared by those who wanted to reestablish the foundation of logarithms from a geometrical basis rather than from exponential equations. Thus the issue proposed is once again the opposition between the two disciplines. In Architettura Civile, however, the most significant fact for the purpose of my argument is that even the theory of the orders, the very core of Vitruvian numerology, is overshadowed by the alternative geometrical approach. Remarkably the modular commodulatio procedure, rooted in numbers, is replaced by a mixed system where the dimensions of the architectural elements are determined by geometrical constructions and only in some cases by numerical operations. Therefore, Guarini breaks away from a long-standing tradition where the only possible way of making the orders had to be numerical.
THE REVIVAL OF EUCLIDISM
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. 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. 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 Elements. Franceso Barozzi edited a renowned edition of Proclus's commentary, in which, as already noted, he acutely observed and stressed the fundamental linear-like essence of geometry. But Barozzi and Barbaro's epistemological interest dwelled upon another important notion, that of "demonstration" (demonstrazione), not coincidentally a basic requisite of the Euclidean axiomatic-deductive procedure. For them, but also for other mathematicians of the Paduan circle such as Giuseppe Moleto as well, the theory (teorica) would have been valid only in conjunction with demonstrations [Tafuri 1985:202]. Barozzi also polemized with Alessandro Piccolomini and Pietro Catena, who argued for the separation of Aristotelian syllogism from mathematical logic, thereby putting the latter on an inferior level. On the other hand, Barozzi in his Opusculum: in quo una Oratio e duo Questiones, altera de Certitude et altera de Medietate Mathematicarum continentur, dedicated to Daniele Barbaro, stressed that "the certitude of mathematics is contained in the syntactic rigor of demonstrations" [Tafuri 1985:206]. To carry this idea into architectural theory was, as is well known, Barbaro's task in his Vitruvian commentary, where syllogism (for Barbaro, discorso) and demonstration are key elements. Therefore not only was geometry at that time compellingly reevaluated but the epistemological value of the geometrical demonstration was appreciated as well, with an interesting architectural twist.
THE DECLINE OF 17TH CENTURY PYTHAGOREAN
To this Kepler replied in the Mysterium Cosmographicum on a geometrical basis. For him the orbs were six because they defined the spaces between the five regular solids. To substantiate the fact that the bodies were five Kepler cited the last proposition of Book XIII of Euclid's Elements. This should not be considered coincidental for, indeed, Euclid was held in the highest consideration by Kepler: for example, in a letter to Heydon in 1605, he writes that the archetype of the world "lies in Geometry, and specifically in the work of Euclid, the thrice-greatest philosopher [et nominatim in Euclide philosopho ter maximo]" [Field 1984:283]. But Kepler's most evident Euclidean concern came out in the field of music, where he tried to fight the Pythagorean conception, exactly in the realm where it was strongest. Kepler's Harmonices Mundi is specially devoted to the founding of musical ratios on geometry. The first book, in which Kepler outlines his theory, is entirely devoted to geometry, the second on music. He declares:
Judith Field has pointed out that "... the weight of the geometrical work in Harmonices Mundi ... must be seen as indicating that he took very seriously his endeavor to prove that God was a Platonic geometer rather than a Pythagorean numerologist" [Field 1984:284]. The case of Kepler further proves that the opposition between Pythagorean theories and Euclidism was a vast phenomenon which transcended the realm of architectural theory. Moreover, Kepler's attitude reveals that the issue, far from involving merely practical procedures, had ontological facets in the deepest sense.
THE CONFLICT BETWEEN EUCLIDISM AND PYTHAGOREAN
In geometry the approach is totally different: the entities adoptedline, 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 linenot 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. Geometry, in fact, often became synonymous with continuous. Mathematicians such as Barozzi, Tartaglia or Vivianijust to quote those from the period with which I have mainly dealtwere 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. 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 tracei.e. the geometical approachor a number to calculatei.e. the numerological approachnot 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 phenomenonof which I have analyzed the revival of Euclidean geometry within Italian scientific circles and Kepler's approach in the fields of astronomy and musicfurther magnifies its importance.
 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 Gromatici and the Regius Manuscript in the King's Library of the British Museum [Euclid 1956, I: 91-95]. return to text
 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: Rara Mathematica]. return to text
 "Proportio est ratae partis membrorum in omni opere totiusque commodulatio, ex qua ratio efficitur symmetriarum," [Vitruvius, III, 1, 1]. return to text
 Girolamo Cardano stigmatizes this opposition when in his De subtilitate contrapposes an "Euclidis Laus," which praises Euclid's "inconcussa dogmatum firmitas," with a rather critical "Vitruvij Laus," where Vitruvius is accused of being only a compiler. See [Oechslin 1983:23]. return to text
 Mario Vegetti has written, "The tradition of the Timaeus remains completely foreign to the theoretic field of the Euclidean-style sciences("La tradizione del Timeo resta del tutto estranea al campo teorico delle scienze di stile Euclideo") [Vegetti 1983: 156]. return to text
 Alberti owned a copy of the Elements. It is now in the Marciana library in Venice. return to text
 Note XVII of Michel Chasles' Aperçu historique ...  has the heading "Sur Maurolico and Guarini". See [Baldini 1980-I; Micheli 1980: 489-490]. On Maurolico see [Clagett 1974] and [Dollo 1979]. return to text
 Unpublished manuscript at the Biblioteque Nationale, Paris. He also translated Euclid's Phenomena. return to text
 On Millet de Chales and 17th century encyclopedism see [Vasoli 1978]. return to text
 "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]. return to text
 "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 ars metendi. return to text
 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]. return to text
 "The diffidence of pure geometry with regards to logarithms" ("la diffidenza del puro geometra nei confronti dei logaritmi.") [Guarini 1968: 418, n. 4]. return to text
 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 Caelestis Mathematica. return to text
 "...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]. return to text
 "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]. return to text
 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. return to text
 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, Mechanicorum Liber (Pesaro, 1577), quoted in [Tafuri 1985:203]. return to text
 To this might be added John Dee's inclusion of architecture among the mathematical arts. return to text
 The connection between syllogism and geometrical reasoning was known since Socrates' times. See [Mueller: 292ff]. return to text
 A good summary is given by [Evans 1957]. See also [Manin 1982]. return to text
 A position like that of Ramus is to this respect symptomatic. On Ramus and French anti-Euclidism see [Bruyere 1984]. return to text
 Guarini gives this topic primary importance. His
Euclides begins with Tractatus I - De quantitate continua
and Tractatus II - De quantitate discreta; these topics
are treated also in several other parts of the book. In Placita
Philosophica one chapter deals with Quantitas and
another with De continui compositione. return
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