HESPERIA 76 (2OO7) IDEA AND VISUALITY Pages SSSS9S IN HELLENISTIC ARCHITECTURE
A Geometric Analysis of Temple A of the Asklepieion at Kos
ABSTRACT
uses The author analytic geometry and AutoCAD software to analyze a the plan of Temple A of the Asklepieion at Kos, revealing circumscribed as Pythagorean triangle the basis for the plans design.This methodology and counter its results earlier doubts about the application of geometry to Doric an temple design and suggest the existence of alternative to the grid-based approach characteristic of Hellenistic temples of the Ionic order. Appre a ciation of the geometric system underlying the plan of Temple A leads to consideration of the role ofvisuality inHellenistic architecture, characterized as manner here the inwhich abstract ideas shared by architects and scholars conditioned viewing and influenced the design process.
on was a The Asklepieion the island of Kos healing sanctuary and medi some cal school of great importance throughout antiquity.1 It lies 4 km southwest of the ancient on a polis of Kos, built terraced slope commanding views sea. impressive of the In its completed state, the complex consisted of terraces three separate connected by stairways, each supporting structures from various periods (Figs. 1,2).2 the of By middle the 3rd century b.c., the sanctuary's three terraces were a constructed.3 On the lower terrace, ?-shaped Doric stoa with ad rooms was to an x m joining built enclose approximately 47 93 space.4Major architectural on terrace an a features the middle included altar, replaced by more monumental version in the following century, and temples dedicated
1.1 wish to thank Andrew Stewart erous of this and see encouragement project opment dating of the site, for his constructive criticisms and for an initial of Schazmann following presentation and Herzog 1932, p. 75; an interest in at an taking my arguments, my arguments Art History and Gruben 1986, pp. 401-410; 2001, which are all the for our con Mediterranean Collo stronger Archaeology pp. 440-449. versations. I am indebted to Fikret at the of 3. Schazmann quium University California, and Herzog 1932, andDiane Favro for their de in All Yeg?l Berkeley, April 2005. drawings pp. 72-75, pis. 37,38. voted attention to this from and are own. 4. on study photographs my A centrally placed propylon its to I am also as inception completion. 2. For the history of the Koan north wing served the monumental to Erich Gruen and Craw see to grateful Asklepieion, Sherwin-White 1978, entrance the sanctuary; Schazmann ford H. Greenewalt for their 345-346. For the devel Jr. gen pp. 340-342, and Herzog 1932, pp. 47-48.
? The American School of Classical Studies at Athens 556 JOHN R. SENSENEY
1. at view of Figure Asklepieion Kos, the middle and lower terraces from
the upper terrace, with the remains of the b.c. of 3rd-century Temple Asklepios (left), the 2nd-century b.c. restoration of the altar (center), and the a.d. 2nd-3rd-century restoration of the Temple of Apollo (right)
to see Asklepios and Apollo (Temples B and C, respectively; Fig. I).5 On a the upper terrace, TT-shaped stoa of timber construction balanced the stoa of the lower terrace.6 b.c. The first half of the 2nd century witnessed changes and additions to the terrace that resulted in a new character as a upper for the sanctuary whole. To connect the upper terrace with the rest of the sanctuary below, a new a a grand staircase created dominant central axis (Fig. 3).7 In addition, new marble stoa the earlier timber structure. In the center the replaced of x m a approximately 50.4 81.5 space enclosed by this stoa, marble Doric was as as as temple of Asklepios begun early 170 b.c., today referred to Tem to ple A (Figs. 2-5) distinguish it from the earlier temple of Asklepios on terrace (Temple B) the below.8 Axially placed before the staircase overlooking the middle and lower terraces, Temple A became the dramatic visual focus of the entire Asklepieion. an The choice of the Doric order for Temple A is archaism. While Doric stoas continued to be common in all areas of the Greek world down
through the Hellenistic period, Asia Minor and the nearby islands reflect
5. Important utilitarian features, 62, 75,98,109,112,149,159,171,246, such as a and wells lo springhouse fig. 74. cated the wall for the 7. Schazmann and along retaining Herzog 1932, were also found on upper terrace, pp. 22-24, fig. 18, pis. 10,11, 37-40, this level. For these features, as well 45-48, 54. as the monumental 8. Schazmann and 2nd-century altar, Herzog 1932, see Schazmann and tem Herzog 1932, pp. 3-13, figs. 3-14, pis. 1-6. The is oriented west pp. 25-31, 34-39, 49-51, 60, 73, ple 25 degrees of north. on a pis. 12-14. Built foundation of limestone, the 6. For the timber and its con portico superstructure of the temple is later marble see replacement, Schaz structed of marble throughout with the mann and courses Herzog 1932, pp. 14-21, exception of of poros limestone naos. figs. 15-17, pi. 9; Coulton 1976, pp. 9, blocks in the interior walls of the AND IN IDEA VISUALITY HELLENISTIC ARCHITECTURE 557
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a predilection for the Ionic order for temple architecture. As Vitruvius as indicates, architects such Pytheos and Hermogenes bolstered this pref erence a as with theoretical justification (Vitr. 4.3.1-2).9 Furthermore, its was measurements demonstrate,10 Temple A traditional in its omission of
to as over 9. In addition Pytheos and the convincing and still much seeTomlinson 1963. common 10. Schazmann and Hermogenes, Vitruvius mentions the looked arguments against the Herzog 1932, a of 2-5. architect Arkesios, who perhaps dates Vitruvian conception of "decline" pp. 3-5, pis. as to the 3rd century. For Arkesios, well the Doric order in the 4th century b.c., 558 JOHN R. SENSENEY
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the kind of novel modifications and "optical refinements" characteristic of the Parthenon. Foregoing also the interesting and easily detectable schemes of Ionic temples associated with Pytheos and Hermogenes (Fig. 6), the seem a temple would to have been strictly conventional reapplication of the Doric order in the 2nd century b.c. Yet the straightforward character of Temple A may represent only a a measure part of its story. As I argue below, geometric analysis of its ments use a reveals the of compass in constructing the interrelationships of architectural elements in plan according to circumferences.11 The di ameters a of these circumferences share simple arithmetical relationship on a based the whole-number proportions of 3:4:5 Pythagorean triangle, amore rather than strictly geometric relationship pertaining to irrational or or numbers like v2 v3, their fractional approximations. The geometry of the temple's plan is therefore very simple, and is not to be confused by the analytic geometry required to substantiate it. The presence of theoretical circumferences concealed within the building's features raises interesting questions about the nature of the on a Doric design process Hellenistic architect's drawing board. That such an a underpinning is found in only single (albeit prominent) example of as to more Greek temple architecture, opposed the widespread approach of grid patterns, does not detract from its significance. As I will discuss, a the uniqueness of circumferential relationships in temple plan?as op to posed the kind of orthogonal relationships that temples of the Ionic a order permit?relates to dearth of specifically Doric temples during the Late Hellenistic period. The interesting geometry in Temple A demon an we strated here exemplifies important architectural tenet that might term to the features of the "cryptomethodic," referring systematic design 11. For an excellent discussion of that cannot be casual but see process appreciated through observation, may plans in ancient architecture, Hasel be 1997. recovered only through detailed study. berger IDEA AND IN VISUALITY HELLENISTIC ARCHITECTURE 559
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IDEALISM AND HELLENISTIC VISUALITY
to a Before turning technical discussion, Iwill first address the very premise an a on that ancient architect should design building based geometry that not does correlate experientially with the final product.12 It is important to state outset from the that architects of the Hellenistic world thought about their buildings in terms different from those used by architects today.We know from Vitruvius that Greek architects called their plan, elevation, and perspective drawings i??ou (Vitr. 1.2.1-2), corresponding to the notion that Platonic idealism uses in reference to the transcendent ideas (or forms) that are to thought be the ultimate reality underlying the perceptible objects of the everyday world. As Lothar Haselberger has admirably observed, the correspondence between the philosophical and architectural meanings is not casual,13 and the full implications of this correlation have yet to be appreciated in studies of ancient architecture. The rhetorical manner inwhich Plato sometimes discusses this ideal can seem our own as ist vision quite foreign to way of thinking, when he can presents Socrates' argument that couches manufactured by artisans only an a our imperfectly imitate archetypical couch existing in realm beyond senses it is to (Resp. 10.596e-597e). Yet unfair reduce Platos conception to these isolated metaphors and parables, and the lack of any clearly stated our unifying theory of ideas in Plato's work should draw attention instead to more as a the general importance of mathematics model for systematic and hierarchical methods of penetrating to the ultimate realities of the most universe in Plato's idealism. Perhaps the articulate expression of this way of understanding is the well-known passage in which Socrates, after an a guiding uneducated slave through geometric proof, concludes that our eternal truths lie beyond embodied experiences in the world (Meno 82b-86c). According to the Platonic model, it is the theoretical rather than the sensory that is privileged. on What brings this discussion to bear the question of underlying terms geometric systems in architectural plans is what J. J. Pollitt the "scholarly mentality" of the Hellenistic age.14 Perhaps originating in the
Roman had no choice but to and based drawn models 12. The following considerations people geometrically not to the ancient and in architecture: "For the transformation pertain only world, look through which they acquired more to how least in their sense of of the ideas into Plato is but generally culturally (at part) subjec measures, I focus from based understandings of the world tivity" (Eisner 2007, p. xvii). helped by analogy practical life, in are not on and what texts where it that all arts and crafts anticipate the way which objects here subjectivity appears a can us are also that viewed and visually constructed. For and images tell about visuality guided by 'ideas,' is, by on the discussion of this idea in the contexts of in the classical world, but rather the 'shapes' of objects, visualized by modern how the of inner of the craftsman who then Cartesian perspectivalism, early geometric underpinning eye A relates to of that them in painting, and 19th-century photogra Temple ways seeing reproduces reality through see 16-17. A were and conditioned. imitation. This enables him phy, Jay 1988, esp. pp. culturally socially analogy to understand the transcendent charac definition of visuality offered by Nor 13. Haselberger 1997, esp. pp. 77, man and and ter of the ideas in the same manner as Bryson (1988, pp. 91-92) has 92-94, primary secondary recently been evoked by Jas Eisner in sources cited. Hannah Arendt (1958, he does the transcendent existence of new a an the which lives the fab his study of visuality in classical p. 90) offers excellent philosophical model, beyond con rication it and therefore context: "the pattern of cultural articulation of how Platonic ideas relate process guides can for structs and social discourses that stand to notions of models and measures, eventually become the standard success or between the retina and the world, a which may be useful for framing the its failure." screen which ... Greek and connection between ideas 14. See Pollitt 13-16. through conceptual 1986, pp. IN IDEA AND VISUALITY HELLENISTIC ARCHITECTURE 561
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Figure 6. Restored plans showing the intellectual ambience of the at a taste for didactic grid systems of Pytheos's Temple of Library Alexandria, of came to characterize Hellenistic Athena Polias, Priene (left), and displays abstruse knowledge strongly of art to Hermogenes' Temple Artemis, and literature. A notable feature of works appears have been the After Coulton common Magnesia (right). 1988, deliberate potential for simultaneous appreciation from both and p. 70, fig. 23 erudite perspectives. In architecture, in particular, this tendency is found in as at examples such Pytheos's Temple of Athena Polias Priene (Fig. 6, left), masses inwhich the might marvel at its surface qualities, while those who as an knew the building's proportions could understand its plan expression of mathematical precision.15 Vitruvius, whose text depends in part upon the writings of earlier Hellenistic architects, exemplifies this scholarly emphasis. He insists that an as architect's background in disciplines like geometry, music, and a at tronomy is requisite (Vitr. 1.1.4, 8-10), claim that he backs up times to with pretentious displays of erudition. Sometimes his eagerness show as his knowledge exceeds his command of the material that he discusses, when he credits Plato with the demonstration of the doubling of the square, which he follows with an introduction to the 15. Pollitt 1986, pp. 14-15. immediately Pythagorean without that both of these theorems illustrate an identical 16. See de Jong 1989, pp. 101 theorem, realizing 102. principle of proportion (Vitr. 9.Praef.4-7).16 Despite such limitations, he R. 562 JOHN SENSENEY
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Figure 7. Plan of a Latin theater demonstrates his in the context not of to Vitruvius's scholarly quality only general theory, according description but also of architectural design. His procedures for designing the plans of both Latin and Greek theaters (Figs. 7,8) well illustrate this tendency, and merit quoting at length. For the Latin theater, he writes (Vitr. 5.6.1-3): to as The plan of the (Latin) theater itself is be constructed follows. a Having fixed upon the principal center, draw line of circumfer ence to at equivalent to what is be the perimeter the bottom. In it inscribe four equilateral triangles at equal distances apart and as touching the boundary line of the circle just the astrologers do in a are figure of the twelve celestial signs when they making computa tions from the musical harmony of the stars. From these triangles, select the one whose side is closest to the scaena and in the spot where it cuts the curvature of the circle let the front of the stage be a set located. Then draw through the center parallel line off from to that position separate the platform of the stage from the space of the orchestra.... The wedges for spectators in the theater should so run be divided that the angles of the triangles that around the cir cumference of the circle may provide the direction for each flight of steps between the sections up to the first curved cross-aisle. Above are to out this, the upper wedges be laid with aisles that alternate at with those below. The angles the bottom that produce the direc seven tions of the flights of steps will be in number, and the remain scaena. ing five angles will determine the arrangement of the In this center to it way the angle in the ought have the "palace doors" facing to and the angles the right and left will designate the position of the doors for chambers." The two outermost will to "guest angles point 17. Smith 2003, pp. 165-166, trans. the passages in the wings.17 S. Kellogg. IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 563
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Figure 8. Plan of aGreek theater And for the Greek theater (Vitr. according to Vitruvius s description 5.7.1-2): some are In Greek theaters things done differently. First, in the bot tom circle, while the Latin theater has four triangles, the Greek has three squares with their angles touching the line of circumference. The limit of the proscenium is determined by the line of the side of the square that is nearest the scaena and cuts off a segment of the cir cle. Parallel to this line and tangent to the outer circumference of the segment, a line is drawn that delineates the front of the scaena. Draw a center line through the of the orchestra and parallel to the direction are of the proscenium. Centers marked where it cuts the circumfer ence to at the right and the left the ends of the half-circle. Then, with the compass fixed at the right, an arc is described from the horizontal at to distance the left the left-hand side of the proscenium. Again, with the center at the left end, an arc is described from the horizontal at distance the right-hand side of the proscenium_Let the ascend ing flights of steps between the wedges of seats, as far up as the first out on curved cross-aisle, be laid lines directly opposite the angles are of the squares. Above the cross-aisle, the other flights laid out between the first. At the top, as often as there is a new cross-aisle, the 18. Smith trans. 2003, pp. 167-169, same number of flights of steps is always increased by the amount.18 S. Kellogg. 19. For the Greek material, see Isler are not to to These prescriptions easy follow, and it would be tempting 1989. For a discussion of questionable dismiss them as a outlook on the of Vitruvius if not to indicating fussy part scholarly attempts match Vitruvius's for the fact that these constructions were in of geometric description Roman theater design applied surviving Greek and Roman theaters.19 The to a basic to later Roman theaters, see Sear 1990; prescriptions pertain geometry 27-29. as or 2006, pp. of forms such equilateral triangles squares, rather than considerations 564 JOHN R. SENSENEY
on case based irrational numerical relationships. In the of both theater types, the cryptomethodic patterns described arguably would not contribute to nor any visible harmonic relationships, would most ancient visitors have are been likely to perceive them. Certainly, there less theoretically grounded ways of determining the locations of radial stairways, boundaries, and seem more on doorways, and itwould sensible to design forms based simple intuition and functional criteria. On the other hand, our very concern with a an these issues may be consequence of inherently modern prerequisite that the design process directly correlate with sensory experience. For the privileged few in the Hellenistic world who could read and understand such passages, there was value of a different kind: the value of discourse. a Furthermore, for material pertaining to world that validates the indepen our own dence of underlying ideas, privileging of the tangible properties in the final built form is arguably misplaced. a Vitruvius's reference to the drawings of astrologers reveals signifi at cant interdisciplinary issue work in such architectural ideas. Given the no reason to scholarly interests of Hellenistic architects, there is believe that to the practice of architectural drawing developed in reference solely the designs of buildings. Another drawn construction that Vitruvius describes was in detail is the Greek avaXrijiua, which the graphic reference for solar declensions that served as the basis for sundials (Vitr. 9.1.1, 9.7.2-7). He an recon provides algorithm for the drawing, which has allowed for its as a struction markedly circumferential design (Fig. 9).20 In this curvilinear the cir quality, the analemma provides intriguing general comparisons with at cumferential geometry underlying the design of Temple A Kos proposed below. Interestingly, Berossos the Chaldean, whom Vitruvius credits with the invention of the semicircular sundial, moved to Kos and established s there a school of astronomy following Alexander the Great conquest of course s Mesopotamia (Vitr. 9.2.1,9.8.1). In the of the 3rd century, Berossos to school amalgamated with elements of the Koan medical school establish moment the discipline of medical astrology, concerned particularly with the as of conception the basis for casting nativities (Vitr. 9.6.2). was I do not argue that there any symbolic connection between Tem some sort The ple A and the analemmay let alone of mystical value. study an we cannot of architectural iconography is inexact science, and lose we a to the sight of the fact that here deal with design that pertains solely s is architect drawing board; it is only the circumferential approach that similar, underscoring the possibility of shared ways of envisioning (and ?izz?ary drawing) forms among architects and those concerned with astral as as phenomena well geometry. As I demonstrate below, the curvilinear not to element in the underpinning of Temple A pertains solar declen a to sions, but rather to Pythagorean triangle. In this aspect, it is similar the ways in which Greeks and Romans began their theaters with squares or an in triangles, and consistent with outlook characteristic of the ways which educated men of the Hellenistic period thought. While the drawn an form the the final built form plan expresses eternal and abstract of idea, not un brings that idea into presence in ways that need readily unveil its mathematical truth to the senses. derlying 20. See, e.g.,Thomas Howes illus In the Hellenistic in architecture was consti period, then, visuality trations in Howe and Rowland 1999, tuted not solely by the perceptions of the casual viewer, but also by the pp. 288-289, figs. 114,115. IN IDEA AND VISUALITY HELLENISTIC ARCHITECTURE 565
9. The analemma to Figure according Vitruvius's description
a epistemological certainty of geometry. Framed by monarchically spon ac sored scholarly agenda and the resulting practices of visualizing form to cording geometry, modes of visual representation established the starting were point for form in disembodied abstractions that subject to math or norms. ematical rules In this way, the squares underlying the placement were of empirical features within the curvilinear Greek theater patently mea real. Similarly, the circles (and the Pythagorean triangle that gives sure to their proportions) underlying the experiential rectilinear forms of at a Temple A Kos discussed below relate to primary consideration: the geometry that defines the visuality of the building by constructing its are eternal idea. The square, circle, triangle, and other shapes the ideas of a a nature that engender the visible things in the world, be they theater, or even a temple, human body (Vitr. 3.1.3). As in Vitruvius's discussion a of theaters, the drawing of building may begin with geometry alone, and only through the process of design arrive at the final form. In further sup port of these observations, I argue that the design for the plan of Temple at a A Kos similarly began with the drawing of Pythagorean triangle, from a which the design and construction evolved into completed expression to that continues reflect its origin, however imperfectly.
QUESTIONABLE METHODOLOGIES
a an The very suggestion of hidden system within architectural plan tends a raw nerve to touch among archaeologists and architectural historians an an alike. Far from striking innovative note, such approach falls squarely a so one within tradition that has tried the patience of readers that itmay day risk outright exclusion from mainstream scholarly research. Before proceeding, it is necessary to briefly address this circumstance. 566 JOHN R. SENSENEY
a on a In penetrating essay the Parthenon, Manolis Korres offers mark assessment to as an edly negative of efforts present that celebrated building expression of ideal numerical relationships and harmonious proportions.21 as notes Characterizing such studies "pseudo-science," Korres the disturb ing tendency to argue theories that contradict the reality of the building. to error in According him, approaches include impudent suggestions of the temple's construction and published measurements; the reliance of proposed on geometric theories inaccurate, small-scale drawings rather than the to degrees of magnitude found in the actual building; the inability credibly correlate the proposed geometric shapes with analytic geometry; and the or even obsessive mystical motivations that may underlie such studies in the first place.22 Observations like these have articulated and, justifiably, even perhaps reinforced general reservations about the rigor and value of in metrological and geometric studies of architecture various periods and locations in the ancient Mediterranean world.23 to Although Korres's remarks may provide salutary caution future stud severe ies, we may not entirely benefit from marginalization of geometric one analysis in Greek temple design. For thing, the Parthenon antedates the use of scale drawings in architectural planning.24 In buildings of the were con Hellenistic period, when such drawings used (Fig. 6), questions more cerning the geometric basis of plans become considerably applicable.25 Writing at the close of the Hellenistic period, Vitruvius (1.2.1-2) clearly terms or creation describes Greek temple design process in of Tragic, the a of quantitative geometric system, and SiaGeGi?, the placement of archi tectural elements according to that established geometry.26 While Vitru vius's comments cannot comprehensively represent Hellenistic practice,
to for details ... but not for the 21. Korres's statements (1994, lowing discussion pertains method resolving issues that of whole unless pp. 79-80) reflect similar misgivings ological carry implications composition buildings at as asWilliam for of ancient were concentric or con going back least far any geometric analysis they partially and not to the centric in Wilson own Bell Dinsmoor (1923a, 1923b). Readers temple architecture, plan." Jones's on Greek architecture of the Parthenon se. of such less familiar with scholarship per acceptance complex geometry not aware of the 22. Korres 79-80. Similar and numerical in circular build architecture might be 1994, pp. systems esteem to Korres criticisms be directed toward the and other structures degree of attached may ings (2000b) has, comes of Greek architectural environ in elicited doubts even and his views, something that study turn, extending across in less formal set ments C. A. Doxiadis in to material of the Roman see, particularly by (1972), period; In an a recent his of the 2001. tings. aside during public cluding analysis Asklepieion e.g., Yeg?l at Kos. both a 24. A forceful lecture at the Art Institute in Chicago, Lacking proper trigono particularly argument referred to metric and identi in favor of detailed architectural draw for example, Jeffrey Hurwit analysis convincing as a In fication of salient architectural features for at least one Korres "genius" (Hurwit 2005). ings Classical-period to his Athenian the is my own view, the remarks of Korres pertaining proposed geometry, building, Propylaia, are characteris Doxiadis that the Dinsmoor 1985. For views referenced in this study suggests sanctuary's Jr. opposing and it is these remarks structures from various eras were in on the introduction and role of drawn tically incisive, that I invoke as a tended to relate to one another in Greek see Hasel and their implications through plans architecture, used lines established at related 83. background for the methodology sight angles berger 1997, p. in of A at Kos. To to the section." He does not 25. For the of scale my analysis Temple "golden development no an of A. See theHellenistic be clear, I in way draw any meaning attempt analysis Temple plans during period ful comparison between Temple A and Doxiadis 1972, esp. pp. 125-126, and alternative modes of architectural 77. in earlier see Coulton the Parthenon. With its markedly fig. design periods, in execution and 23. In of his for 51-67. greater sophistication support theory 1988, pp. an "facade-driven" Doric in the 5th 26. For from details, the Parthenon is expression design complexities arising a Mark Wilson Vitruvius's use of Greek of completely different mentality century b.c., Jones (2001, terminology we in and is advocates "a rule in this see Fr?zouls from what find Temple A, p. 678) general [that] passage, 1985, esp. a different era. The fol ancient architects 217. the product of exploited geometry p. IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 567
con his stated reliance upon Greek architectural writers arguably merits tinued investigations into the geometric underpinnings of Hellenistic buildings. In addition, Vitruvius's apparent adherence to grid-based ap proaches in Ionic temple design27 might elicit inquiry into procedures that he does not elaborate upon: in the Doric order, where intercolumnar spatial not to an contractions do lend themselves orthogonal grid, how might the geometric constructions of taxis differ?28 a As privileged monument, furthermore, the Parthenon will continue a numerous to be favored object of attention for lines of inquiry despite condemnations of particular approaches. Dating from the 19th century are re onward, however, too many volumes of published archaeological we ports with scientific measurements pertaining to buildings about which or still know relatively little. To allow these to gather dust occupy unused electronic storage space, instead of reaping what the laudable efforts of excavators can us their tell about ancient architectural design, will benefit nor accusa neither archaeologists architectural historians. Furthermore, tions of "intellectual totalitarianism"29 directed at proponents of geometric serve to analysis could only curtail productive discussion. as or as ra Rather than framing various outlooks scientific mystical, or we see as tional obsessive, might instead observations such those of as an to Korres opportunity reevaluate the methodologies employed in an proportional and geometric analyses. In addition, inclusive view may us open to methods that allow for scientifically sound analyses that, in turn, our a solidify understanding of Hellenistic temples. Finally, responsible, mathematically rigorous, and computer-based approach to geometric us analysis will help build upon and refine the criticisms, rules, and expec tations of similar studies. uses The present study analytic geometry and vector-based AutoCAD to (or CAD) software analyze the geometric underpinning of the design of at course Temple A Kos. In the of this analysis, I also consider questions surrounding the perceived limitations of studies that attempt to unveil hidden numerical and geometric systems. In order to avoid the inevitable correct distortions of proportional and geometric relationships that look on a to only when overlaid plan drawn reduced scale, my study instead on directly relies the buildings published measurements. In other words, now the proposed geometric system is mathematically verifiable rather than we intuitive, and is grounded in computation. So that may furthermore ensure nu 27. See the comments of Thomas both mathematical accuracy and the relationship between the Howe in Howe and Rowland 1999, merical systems and the concrete, graphic form of the revealed geometry, pp. 5,14,149. use the calculations have been verified through the of AutoCAD. CAD 28. See Vitr. 4.3.1-8, where he not a a is requisite for this study, but merely convenient tool that may expresses his indebtedness to the Ionic allow researchers and readers a recourse to the measurements of of simpler tradition Hermogenes by charac as in an architectural it is the calcula terizing the Doric order deficient, proposed relationships form; ultimately leaves the issue of columnar interaxes tions themselves that demonstrate the geometry. This combined Cartesian and focuses on elevations unexplained, and computer-based method carries the potential of standardization for at the expense of any discussion of a future studies, allowing for truly scientific approach inwhich results may Wilson related notion of plans. Jones's to an as be replicated confirm their veracity. Provided that analysis such "facade-driven" Doric design (2000b, this one relies numbers rather than ones own pp. 64-65; 2001) is discussed below; see upon previously published we now set aside of and also n. 23, above. measurements, may suspicions personal agenda 29. Korres 1994, p. 80. have confidence in the objectivity of the process. R. 568 JOHN SENSENEY
ANALYSIS OF THE BUILDING
The Metrology
Before discussing the nonorthogonal dimensions revealed through analy we s sis, should consider the temple general measurements. From the 5th was a common century onward, it rule of thumb that the width:length a ratio of Doric temples plan (including the euthynteria) should match on the number of columns the short and long sides of its peristyle.30 With rear six columns along its front and and eleven along its flanks, Temple A no appears to be exception (Fig. 10). In plan, the temple's overall dimen are x a a sions 18.075 33.280 m,31 differential of only 0.4% from proper as a m 6:11 ratio. A simple adjustment, such 0.143 reduction of the overall or a m a length 0.078 increase in the width, would result in perfect whole number ratio.
Scholars usually account for such "errors" by citing constructional in as as exactitude and adjustments, well centuries of exposure to the ele ments.32 Other slight irregularities found throughout the temple might support this notion of a difference between the theoretical design and are the actual built form (Fig. 5). For example, there slight variances in the thickness of the eastern and western naos walls (1.028 and 1.016 m, to respectively) and in the distances from the exterior of these walls the a edges of the stylobate (3.313 and 3.380 m, respectively).33 As result, the naos on is not centered the stylobate.34 as over Although factors such imperfect masonry and deterioration are time plausible explanations for such disparities, additional consid to a erations deserve emphasis. If it is the architect's design begin with proper 6:11 plan, other features might complicate the maintenance of as perfect proportions in the final built form the construction progresses. a In the end, there will be set of measurements that are necessarily inter as related, such the widths of the krepidoma and the overall dimensions of
m on 30. See Coulton 1974, pp. 62-69; on the eastern side runs in courses that is 0.067 less than that found are with of the western side. This is Wilson Jones 2001, p. 694. that parallel the long walls interpretation on the western runs the distances 31. Schazmann and Herzog 1932, the naos, that side supported by nearly equal in courses that are of 4.43 and 4.435 m from the naos pi. 2. roughly perpendic to walls to the outer of the 32. For the problem of the differ ular these walls. In addition, the edge euthyn ence on western are teria on the western and southern between the abstract vision of joints the side tighter sides, see east. as to 4.368 m on the eastern the architect and the final product, than those toward the These opposed into side. If we maintain that Wilson Jones 2000b, pp. 11-14; Dwyer divergent tendencies continue the approximately naos m was 2001, p. 340. raised foundations of the itself, 4.435 originally intended for two the eastern side as the result 33. Schazmann and Herzog 1932, where the separate approaches well, 2. meet at a line west of the central axis would be a more balanced than pi. design excavators A of the naos. if the entire celia on the 34. The of Temple originally lay a in its reason that this lack of symmetry is The difference should indicate that central axis of the temple present shifted crews were for dimensions. I therefore favor human result of earthquakes that have separate responsible lay naos on error as to natural causes for the entire and pronaos eastward, ing the limestone foundations either opposed an that I find unconvinc side of the More than the lack of in the explanation temple. plausible symmetry temple's see the eastward shift of the entire celia is measurements. Such errors can and do ing; Schazmann and Herzog 1932, on on-site there that one crew committed a minor error occur in the of p. 6. Based my analysis, laying foundations, are in eastern limit of the the of elements in differences in the limestone foun establishing the affecting placement on eastern western or the dations the and stylobate, possibly the euthynteria, superstructure. naos. in a from sides of the While the masonry resulting distance the celia IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 569
14.666 M = 45 F = 24 T
4^
o i-4 U h
10m
Figure 10. Restored plan of Temple A, with measurements of the colonnade axes = = (M meters, F Doric feet, = T triglyph width modules) 57? JOHN R. SENSENEY
turn to the the stylobate, which in relate the sizes of the paving slabs and area spacing of the columns they support. The where the relative propor tions of the various parts are subject to modification is the conversion from as x to the abstract units of the drawing board (such 6 11) actual metric must values. In determining specifications, certain distances be privileged while others must be adjusted to the space allotted them. Considerations as such the specific measurements of the paving slabs, for example, may a ultimately result in slight departure from the integral proportions of the s architect original drawn plan. One method of accounting for the overall and individual dimensions a a recent the of building is metrological analysis. A study proposes that out architect of Temple A first worked the overall dimensions according a were corner to specific metrological system.35 Only thereafter the usual to contractions of the Doric order worked out, resulting in adjustments the dimensions of the theoretical plan. This theory, however, relies upon m as common the identification of a 0.305 "foot" the unit underlying the exists no such unit of temple's metrological system. Simply put, there measurement in the ancient Greek world, a fact that the theory's authors in our of contend with by advocating greater flexibility understanding Greek metrology.36 new we Instead of suggesting units of measurement, may consider the issue of commensuration. For Doric temples, specifically, Wilson Jones makes a detailed case for amodular system, at least for 5th-century exam to a ples.37 According this theory, the width of standard triglyph expresses the module that establishes commensurability throughout various elements to a of the building.38 The triglyph module itself commonly corresponds a or with a to 5-dactyl multiple of standard foot (e.g., 25 30), dactyl equal 1/16 of a foot in accordance with Greek metrological standards.39 measurements in situ are available for the In Temple A, for the remains central columnar interaxis and the western half of the columns on the rear as as western lateral colonnade of the stylobate, well four columns along the m eastern half (Fig. 5). At the rear, the addition of 5.793 for the missing + + a m (5.793 3.080 5.793 m) results in length of 14.666 for the entire excavators columnar axis (Fig. 10). For the long sides, the temple's posit interaxes of 3.05 m based on the remains in situ and a consideration of measure 0.61 and 0.915 the triglyphs and metopes, which m, respec one of 3.034 m tively (Fig. II).40 Thus the preserved interaxis (see Fig. 5) an constant would represent unintended departure from the theoretical of 3.05 m, and we may thereby restore the theoretical lateral axes, excluding to a m x as in 10. the contracted corners, length of 24.4 (8 3.05 m), Figure m Therefore, the 24.4 m axes of the lateral colonnades and the 14.666 axes of the front/rear colonnades would equal 40 and 24 integral units, a to 0.61 m respectively, of value equal (Fig. 10).41
35. Petit and De Waele 1998. 37.Wilson Jones 2001. Regarding 40. These measurements are based such a could on three of the 36. Petit and De Waele 1998, esp. the possibility that system surviving fragments an have endured into later see the see Schazmann and p. 62. In earlier essay, J. J. de Jong periods, frieze; Herzog to measure author's comments on n. 107 10-11. claims have analyzed the p. 697, (in 1932, pp. = = ments but offers no dis to Coulton 41.24.4/40 0.610 14.666/24 of Temple A, response 1983). m; or to his anal 38.Wilson 2001. 0.611 m. cussion results pertaining Jones see 3. 39. Wilson 690. ysis; de Jong 1989, esp. p. 104, fig. Jones 2001, esp. p. IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 571
.345fr
.960 v 1.5r
* ? .610 * .803
\\\\ \ MI IM I M Hl <--1.056-5>j
3.05 = 5 T
HS?1.515-3H 0 5 m Figure 11. Restored elevation of lateral colonnade of A, with Temple As widths also measure 0.61 m.42 A measurements in meters and Significantly, Temple triglyph triglyph calculation shows that this value 30 of a 0.325 m width modules (T) simple equals dactyls a "Doric" foot.43 This triglyph width shares 2:3 relationship with the a standard metope, 1:5 ratio with the average interaxial spacing of the lateral colonnades, and a 1:24 ratio with the axis of the facade colonnade? to all typical proportional relationships according Wilson Jones's study (Figs. 10, ll).44 It is also interesting that these distances of 40 and 24 42. Schazmann and combined with measurements taken to measurements Herzog 1932, according the of 10-11. from various on pp. buildings the Acropolis Ifigenia Dekoulakou-Sideris (1990), 43.0.610 m/30 = 0.02033 m. Since a and Attica. The now throughout investiga has been convincingly shown by foot divides into 16 dactyls, 0.02033 m tions ofWilliam Bell Dinsmoor (1961), Wilson Jones (2000a) to represent a x 16 = m. between coined term con on a m 0.325 Varying 0.325 who the Doric foot, system based 0.3275 to 0.3280 and 0.329 m, the Doric foot has been firmed a value of 0.326 m. See also Doric foot. For the divisibility of known since Wilhelm D?rpfeld's study Wilson Jones 2000a, p. 75; 2001, the triglyph module into 20,25, 30, of the 689. The relief from see (1890) late-5th-century inscrip p. metrological etc., dactyls, Wilson Jones 2001, tion in to relating the expenses involved Salamis, previously thought repre p. 690. a on a m the construction of the Erechtheion, sent system based 0.322 foot 44.Wilson Jones 2001. 572 JOHN R. SENSENEY triglyph modules correspond to precisely 75 and 45 Doric feet.45 In translating the drawn plan to the actual dimensions of the building and its features, then, it is reasonable to theorize that the architect may have axes privileged the colonnades of the facade and rear, establishing their a of 45 Doric feet. Through this magnitude, 3:5 ratio finds the 75-foot axes measurement for the of lateral colonnades (Fig. 10). This latter di mension divides into eight intercolumniations, each of which subdivides one two into two half triglyphs, whole triglyph, and metopes (Fig. 11). axes es Furthermore, the distance separating the end columns of these tablishes the measurements for the contracted corners, and the remaining three interaxial distances of the facade and rear colonnades could be set according to the criterion of incremental widening toward the center. In varying the dimensions of the individual paving slabs in accordance with are es this irregular column spacing, the total dimensions of the stylobate are tablished, and the widths of the stereobate and euthynteria set according over to the remaining distance necessary to maintain the 6:11 ratio of the all plan. to treat as we To insist upon this explanation, however, is Temple A one more have the Canon of Polykleitos, resulting in yet plausible theory that can never be proven. There are too many types of metrical units, too us to many ways of measuring, and too many rationales for induce conclu a sively guiding metrological system. What is lacking in such approaches so a a is not much reasonable correspondence to pattern of numbers, such as whole-number ratios, but rather something outside of the buildings as a themselves that might verify the significance of those numbers, such source or a primary basis in Euclidian geometry. While Vitruvius validates case the relevance of the triglyph module, the for how this system relates we to large-scale distances must remain provisional; in this regard, may wonder, for example, why the 40 integral units of the lateral colonnades as exclude the corner interaxials. In addition, the modular theory applied a to Temple A cannot address central aspect of design that is unrelated naos to the trabeation: the placements of the walls of the and pronaos in must relation to the overall plan. We therefore explore other methods of a analysis in seeking to substantiate theory for the underlying logic of the buildings design. The Theoretical Plan and Standards for Accuracy not to As Korres emphasizes, it is enough merely draw geometric shapes over a to a the features of plan reduced scale of 1:100.46 Instead, proposed must other geometric shapes be verified through analytic geometry. In a to words, superimposed drawing should correspond the elements they not co overlap only visually, but also mathematically through Cartesian 45.14.666 m is 0.026 m ordinates with interrelationships expressed algebraically, and with lines only (or 0.18%) in excess of 14.640 m. described in terms of and curves with coefficient-based formulas, slopes 14.640/24 = 0.610 m; 14.640/45 = for such a strict standard a on continued example. Naturally, places damper 0.325 m; 24.4/40 = 0.610 m; 24.4/75 = attempts to theorize about ancient architectural plans, but the gains in 0.325 m. are 46. Korres 80. credibility arguably well worth the endeavor. 1994, p. IN IDEA AND VISUALITY HELLENISTIC ARCHITECTURE 573 a Another point of emphasis has been that the degree of accuracy in must in plan's theoretical geometry approximate the tolerances the actual construction.47 Determining the accuracy of the built form, however, elicits a bit of circular reasoning, since many of its elements must be measured same to against the very theoretical plan that its author attempts support.48 case ex This need not be the for every feature, however. In Temple A, for ample, there exist slight variances in the noncontracted columnar interaxes m are of the lateral colonnades, such as 3.050 and 3.034 m,49 that likely to relate to a theoretical constant rather than an intentional irregularity. a we note naos m On larger scale, may that Temple As (9.272 wide, not exact center m including its walls) lies in the of the stylobate (15.965 an m wide), but imperceptible 0.067 off axis (Fig. 5).50 Given the gen even eral predilection for symmetry in conjunction with "optical refine one as ments," would be hard-pressed to argue the plausibility of this feature intentional. From the outer wall to the edge of the stylobate, the dis tances on the western and southern sides of the naos measure 3.380 and measurement m on 3.375 m, respectively, and the diverging of 3.313 the eastern side an error of ca. represents 1.98%.51 Still, it may be inadvisable to isolate this error in the eastern pteron, since the final built form is the product of multiple interrelating compo nents. The most conservative approach would be to calculate the percentage to of tolerance according the entire width of the stylobate. This calculation should pertain to the theoretical plan rather than the actual plan, with the m eastern only difference being the addition of the "missing" 0.067 from the a m side of the temple, resulting in width of 16.032 for the stylobate and an m overall width of 18.142 (see Appendix 1). In order to maintain the strictest possible tolerance in my analysis of this theoretical plan, I will cap the standard for accuracy at 0.42% in accordance with the divergence discussed here.52 It is important to emphasize that this addition to the width in the theoretical plan is slight, and does not in any way "stack the deck" for the m nar results of the analysis that follows. Instead of adding 0.067 to the we rower side, may be justified in adjusting for symmetry in the theoretical naos to plan either by maintaining the actual width and shifting the the or naos center (see Appendix 2),53 by reducing the width of the by 0.067 m in order to balance the sides evenly (see Appendix 3). As the calcula tions provided inAppendixes 2 and 3 demonstrate, the results for each of these alternative theoretical plans remain well under the strict tolerance case theorists are con surements see 47. Korres 1994, p. 79. such incapable of and others, Fig. 5, and 48. In 79 Schazmann and 2. Korres's words (1994, pp. ceiving)." Herzog 1932, pi. to 49. Schazmann and m = 80), theorists "refuse be bound by the Herzog 1932, 52. 0.067/16.032 0.42%. 2. methodological requirement that the pi. 53. This solution would be consis to a 50. Schazmann and tent the views exca degree which theoretical definition Herzog 1932, with of Temple A's or 2. For this and all of the follow who the as (whether metrological, geometric, p. 6, pi. vators, explain displacement to measurements of the naos and the of an whatever) approximates the actual ing pro result earthquake that shifted no dimensions relate to the outside entire see building should be less than the de naos, the celia; Schazmann and the of the walls rather than the socle. 6. gree of accuracy with which build plane Herzog 1932, p. For the problems itself was in 51. 0.067/3.380 m. For these mea with see n. ing constructed (which any this theory, 34, above. 574 JOHN R. SENSENEY to case of 0.42%, and in fact produce results closer 0% in the of several dimensions. The rationale for privileging the theoretical plan inAppendix 1, not to most to therefore, is provide the convincing analysis, but rather adjust a for symmetry in way that most thoroughly relates to the measurements of m to the actual plan; when 0.067 is added the eastern side of the stylobate, one as as the eastern and western sides of the plan equal another well the side behind the southern wall of the naos.54 x m one The theoretical plan of 18.142 33.280 solves problem but one our leaves another unresolved. On the hand, expectation for integral proportions in the overall plan is satisfied, since the theoretical plan results a in nearly perfect 6:11 form.55 On the other hand, the rationale for the naos placement of the and pronaos remains unclear. While the distance of naos an the walls of the from the edge of the euthynteria maintains equal 1:1:1 ratio on the sides and rear, the space before the antae of the pronaos no we shares integral relationship with these distances.56 Nor may readily discern any meaningful proportional relationship in the length-to-width naos dimensions of the and pronaos.57 As I argue below, this lack of ob a servable correspondences pertains to process of design grounded not in a arithmetical relationships between orthogonal dimensions, but rather to geometric procedure executed with the rule and compass. This geometry some to is quite simple, though it requires detail and rigor substantiate we recover it. In the following section, I demonstrate how may the plan's specific design process through analytic geometry. Geometric Analysis on on To properly analyze the plan, I rely simple calculations based the a published measurements of Temple A, with the only adjustment being on m centered naos, flanked either side by equal distances of 0.380 from naos the outer walls of the to the edges of the stylobate.58 All relevant di are agonal relationships in the plan mathematically verified and expressed a a in the footnotes with reference to single quadrant of two-dimensional coordinate system. In addition, Appendixes 1-3 with accompanying Fig ure 23 provide magnitudes, equations, and tolerances that demonstrate the measurements proposed geometry according to for all three theoretical plans described in the prior section. Whenever relevant, the location of features corner will be given as Cartesian coordinates, inwhich the southeastern of the euthynteria's outer edge is at the origin 0,0, and the extreme northwest 54. The southern side measures 56. The distance from the pronaos 57. The overall dimensions of the 4.430 m from the exterior face of the to were it naos and are 9.272 x 22.053 m. the stylobate edge, preserved pronaos naos to outer on ratio is wall of the the edge of the the northern facade, would be Here, the closest integral 3:7, is 5.742 m. from the whose tolerance of 1.9% is un euthynteria, which essentially equal The distance pro again to measurement of west naos to outer the 4.435 the the edge of the euthyn acceptable. ern side (see Fig. 5). Adding 0.067 m to teria is 6.797 m. Of these two measure 58. Thus, the distances between narrower eastern can the exterior walls of the naos and each the side of the stylo ments, the closest integral ratio I a 1:1:1 is a 2:3 the outer of the bate, therefore, produces nearly find relationship between long edge euthynteria equal rear to rather than the 4.368 ratio for all three sides. lateral and distances the euthyn 4.435 m, present = see n. 55. (18.142/6) x 11 33.260, a dif teria (4.435 m) and that of the front and 4.435 m; Fig. 5. See also 34, 0.02 m the an toler above. ference of only from plan's (6.797 m), with implausible 33.280 m length. ance of 2.1%. IN ARCHITECTURE IDEA AND VISUALITY HELLENISTIC 575 m I II Figure 12. Restored theoretical plan M,M,M,UP of Temple A with geometric under pinning corner to at 18.142,33.280. In addition consulting the calculations provided here, readers may replicate the proposed findings using CAD software. The were results of the following analysis verified with AutoCAD.59 a A significant result of this analysis emerges from the location of theoretical central point from which the outer corners of the antae and corners are the outer back of the euthynteria equidistant, and thereby a share theoretical circumference (Fig. 12).60 The pertinence of this cir cumference to the design process is supported by the rational relationships a it shares with other features. The overall width of the temple shares whole-number 3:5 ratio with the diameter of the theoretical circumfer ence, with a tolerance of less than 0.1%.61 If caution advises us to consider a an this ratio possibly fortuitous result, there is additional whole-number on the central a theoretical line to m + a 59. For consistency, all magnitudes axis, figures (15.123 15.124 m) finds are rounded to the millimeter. the external corner of either anta mea theoretical diameter of 30.247 m; see 60. a on the sures 15.124 m. the exter From point located Specifically, Appendix 1. at corner of western anta is at x = plan's long central axis [9.071, nal the 61. (30.247 m/5) 3 18.148 m, a to a mm 12.101], theoretical line [0, 0] [13.749,26.483]. Through simple sub difference of only 6 from the measures 15.123 which is the we find these coordinates at m, square traction, plans overall width of 18.142 m, and m m root of the sum of the squares of 9.071 distances of 4.678 and 14.382 therefore a tolerance of less than 0.1%; and 12.101. From the same coordinates from [9.071,12.101]. The sum of these seeAppendix 1. 576 JOHN R. SENSENEY X/ I-' _ ' r ~ _ r .... __ 0__ _--_ <'^) ~~~10m 0 13. Restored theoretical our Figure plan proportion that should give pause to skepticism: the distance from of Temple A with geometric under the theoretical circumcenter to the southern and the overall plans edge pinning a a width of the temple share 2:3 ratio, again with tolerance of less than a x-x 0.1%.62 We may illustrate this correspondence with baseline of 62. x 2 = 12.095 m, 3 units drawn across the entire width of at the ordinate (18.142 m/3) plan correspond a difference of 6 mm from the to the theoretical center of the with a line only ing point circumference, along ordinate at 12.101 m, and therefore a y-y of 2 units drawn from the circumcenter to the edge of the euthynteria tolerance of less than 0.1%. (Fig. 12). 63. The theoretical diameter of the we circumference 30.247 m Geometrically, may express this relationship through the algorithm larger equals (see n. 60, above), which has a ratio of two circumferences with a radius of 2 units, each centered on either of 5:4 with 24.202 m (the diameter cor terminus of baseline x-x The which is (Fig. 13). larger circumference, to the radius of 12.101 m in responding centered at the middle of intersects with the smaller circumferences x-x x-x\ the y dimension from the baseline at the of the outer corners of the Both the to corner of the exactly points euthynteria. either back euthynteria are at and with an error mathematical proof for and significance of these intersecting points [0, 0] [18.142, 0]), of less than 0.1% calculated the revealed by the whole-number ratio of the diameters of the smaller and by difference divided by the magnitude: circles, 4:5 with a tolerance of less than 0.1%.63 When = larger equaling (30.247 m/5) x 4 24.198 m, a dif conceived in relation to the overall width of the 3 units of temple (the ference of 4 mm from 24.202 m. = x-x), this final dimension brings the geometric principle underlying (24.202 m/4) x 5 30.253 m, a differ a ence of 6 mm from 30.247 m. the architect's system into striking clarity: the 3:4:5 dimensions of IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 577 X -0-- 0 0 _ _ Os__ _ Figure 14. Restored theoretical plan of Temple A with geometric under K^)0~~~ 1~~~~~010m pinning circumscribed Pythagorean triangle.64 In effect, this geometric form ABC at lies the heart of the design, with the compass centered midway along its hypotenuse and the circumference coinciding with its angles and lines (Fig. 14). We should understand this geometric underpinning and its compass as based construction interdependent. Even in the Roman period, architects did not work with a square, let alone a T square. Instead, the method of a producing perpendicular lines with the highest precision employed rule and compass, with straight lines drawn through circumferential intersec same manner tions in the that is revealed through this analysis of Tem A.65 It as ple has already been observed that Roman buildings such amphi a theaters would commonly begin with Pythagorean triangle, and arrive at the final design using the compass through various stages.66 This Roman use a of the Pythagorean triangle recalls conceptually similar manner of 64. The theoretical diameter of the to m in corresponding the radius of 12.101 any dimension of less than 0.1%. m larger circumference equals 30.247 in the y dimension from the baseline 65. See Roth Conges 1996, pp. 370 n. The of x-x to either corner (see 60, above). magnitudes back of the euthyn 372; Taylor 2003, p. 38. m or 18.142 (the plan's overall width, teria at [0,0] and [18.142,0]), and 66.Wilson Jones 1993, pp. 401 baseline 24.202 m diameter 30.247 m = with a maximum error x-x), (the 3:4:5, 406, figs. 13,15,16. 57? JOHN R. SENSENEY ------l ------ Figure 15. Proposed general con struction of a 3:4:5 Pythagorean to circumferential triangle according with dashes intersections, indicating the baseline was constructing Ionic column bases that familiar to Greek architects as as early the Archaic period.67 The transparency of the plan of Temple A us a may allow to understand how Hellenistic architect might construct the a Pythagorean triangle itself. The formula appears to consist of baseline of 6 units, upon the center of which a compass with a radius of 5 units is set, on and the ends of which are set compasses with radii of 4 units. By these to means, the intersections could be joined form the perpendicular lines as as of the triangle's sides well the diagonal of its hypotenuse (Fig. 15). case In the of Temple A, it appears that the larger circumference of this to geometric construct remained in place define the extent of the pronaos at the antae (Fig. 14). now us a To dismiss these results would require to posit confluence of three separate coincidences of whole-number proportions (3:4:5) with a maximum error that is consistent with the strictest possible standard a of tolerance observable in the actual building, along with fourth (and more an conspicuous) coincidence that these proportions engender inte gral geometric form of central significance to Greek mathematics. More a over, the circumscription of Pythagorean triangle graphically expresses at a Tha?es' theorem: three perpendicular bisectors meet circumcenter on runs located the hypotenuse, which the length of the circle's diameter 67. For the Pythagorean triangle In turn, the basic that the (Fig. 16).68 proportions Pythagorean triangle and column bases, see Gruben 1963, establish the location of the theoretical center and the di yields point pp. 126-129. ameters 68. a of the circumferences (Fig. 15). In the face of these internal By definition, Pythagorean to is a see Eue. correspondences and their pertinence Euclidian geometry, the balance triangle right triangle; Elem. this form falls on the side of in 3.31. concerning resulting obviously heavily 69.Width and of naos tentional rather than chance. length design its 9.272 and including walls equal There is another that the to yet integral proportion completes geometric 15.572 m, respectively. According of the The across the naos from the we underpinning temple's plan. diagonal Pythagorean theorem, then, corner corner a each and find the root to including its external walls shares 1:1 correspondence square square m. a of their sum, thus finding 18.123 with the total width of the temple, with difference of only 0.1%.69 From Ifwe take the 0.019 m difference a theoretical central located on the cross-axes of the naos, therefore, point between 18.123 and 18.142 m (the the distance to either of the width and each of the external edge temple's total adjusted width of the temple) and corners of the naos is This a circum essentially equal. congruency suggests divide by either 18.123 or 18.142 m, we a a ferential underpinning to the design of the naos, whose diameter shares find difference of 0.1%. IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 579 010 Q 0 ---=- --S------ X -c -~ '-0^~~ - :- = - L d m "S^~~~loZZ m 16. Restored theoretical Figure plan 0 10~~~~~~~~~~~~~~~~~~~1 0 0t of Temple A with geometric under pinning a whole-number 3:5 ratio with the diameter of the large circle, with toler ance we as a of 0.1% (Fig. 17).70 If accept these circumferences guiding method for the placement of features within the plan, their ratio would call s to mind Vitruvius formula of 3:5 circumferences for the main proportions of plans in peripteral round temples (Vitr. 4.8.2). One indication that the circumferences here suggest an intentional geometric underpinning is their planar interrelationship. The distance separating the ordinates of their theoretical center points is 0.115 m.71 In a are scale plan, special markings required to make this separation percep tible (Fig. 18). Before considering why the architect might have centered at a s we his compass different points hair width apart in his design, might ac consider this separation in relation to the theoretical proportions and tual dimensions to which it corresponds: the separation of these ordinates 70. x 3 = 18.148 n. For outer (30.247 m/5) m, [9.071,12.101] (see 60, above). edge of the euthynteria: (15.572 a + difference of 0.1% from 18.123 m; the smaller circle, the coordinates of the m/2) 4.43 m. For the distance sepa x = a center are center (18.123 m/3) 5 30.205 m, differ point [9.071,12.216]. The rating the theoretical points of - ence of 0.1% from 30.247 m. ordinate here is determined the sum by the larger and smaller circles: 12.216 71. The coordinates of the theoreti center naos = m. of the point of the and the 12.101 0.115 center are naos cal point of the larger circle distance of the from the south 580 JOHN R. SENSENEY ------0 ------\ _ = 17. ^= Figure Restored theoretical plan Y-. of Temple A with geometric under BiaS~~iiiiff~~ia1Q lo pinning represents a 0.38% difference, sowe remain within the strictest standard for theoretical tolerances of 0.42% calculated according to the constructional inexactitude found in the actual building.72 On the other hand, the applicability of this standard here is dubious. we cannot Although conclusively determine the precise metrological sys tem underlying Temple A, the fact remains that the architect or builders to convert would have needed any conceptual circumferential geometry to actual measurements for orthogonal distances. After all, we cannot ex masons to out pect have laid the building according to invisible circles with an to a eye maintaining shared theoretical center point. Due to such nec essary adjustments in the planning and building process, it is natural that deviations from elements are bound to occur. we original design Since 72. As in the calculation of error lack secure access to this of metric the to m intermediary stage specification, relating the difference of 0.67 in a the widths of the in rela relevance of precise calculation for the percentage of error in a common ptera (0.42% tion to the entire width of the center point (such as 0.38%) may be limited. Instead, we may conceive of stylo the difference is here the inmore terms: in a over 33 m we bate), calculated divergence experiential building long, to the find the two theoretical circumcenters of the cir according complete geometry, integrally proportioned as represented by the diameter of cumferences at 0.115 m or less than the of a small points only apart, length the larger theoretical circumference: child's hand in relation to the distance from floor to vaults in the cathedral 0.115/30.247 m = 0.38%. IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 581 1 zzzzzz ?-DII ~-Q-) I --Q" \>QZ EEEEz / oizzzzzzziq-; 18. Restored theoretical Figure plan * i i i i i i i i of Temple A with geometric under iii i^~^ i and indicators the pinning marking 10m 0 circumcenters fc\7) a even of Notre Dame in Paris. In structure where the width of the stylo an cm an bate is off by 6.7 cm, additional inexactitude of 4.8 for invisible was no feature is insignificant, particularly when that feature longer relevant during the actual building process. The Schematic Plan Leaving aside the security that mathematical justification affords, I will now to suggest possible ways inwhich this geometry relates the Hellenistic s architect process of designing Temple A. Unlike the theoretical demon stration above, the following analysis takes into account the proportions to of the actual building. My intention here is explore further questions relating to the design process, integrating what I hope is well-grounded speculation with the results of the above geometric analysis. to In designing Temple A, the architect would have needed harmo naos nize the 3:4:5 triangle underlying the placement of the and pronaos a with the 6:11 ratio of the overall plan. Keeping in mind how compass is centered, it isworth emphasizing that the simplest way of working with the tool is to conceptualize circumferences in terms of radii rather than 582 JOHN R. SENSENEY ' i i i ; i '; i ; r ' i ; i i, ?'i K')l I?I tV I?I? Figure 19. Restored theoretical plan of A overlaid with intersec diameters. In this way, one need not resort to half-number divisors, such Temple tions of circumferences with radii of as 2.5, in order to create a whole-number diameter such as 5 units. In on 3 units producing radii of 3, 4, and 5 a baseline of 6, therefore, the architect a would have created 6:8:10-unit triangle. Extending this same divisor to an the overall plan, additional 3 units in the y dimension produces the final 6:11 ratio of the temples plan, which repeats the 6 x 11 number of columns for the intended colonnades and simplifies the process of drawing by maintaining integers. This s demonstration of the architect method of locating the wall termini to not according circumferences still does explain the rationale behind were where they placed along those circumferences. It is tempting to a suggest simple circumference-based algorithm whereby the architect out might have worked these placements. On the baseline x-x of 6 units, center the on the termini compass and center, drawing three circles of equal radius. Repeat this procedure three times, each with radii of 3, 4, and 5 corners units, finding the location of the walls and according to the cir cumferential intersections (Figs. 19-21). Despite the appeal of the resulting plans, however, itwould be inadvisable to adopt this procedure. As Korres we cannot on recognizes, draw conclusions concerning geometry the basis of how that geometry appears to coincide with features when overlaid on a scale we must such 73. plan.73 Rather, replicate results mathematically. Unlike Korres 1994, pp. 79-80. IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 583 -NIo-- / C-:o \s - _ /111.~-^ / ~ I Ii I 1_1... ..' . .. 0 - - \ / I. . I . I ? ~lo10m 0^J)~~ 20. Restored theoretical case Figure plan the of the underlying geometry demonstrated above, calculations do of A overlaid with intersec a Temple not verify the hypothesis suggested here in way that satisfies the strictest tions of circumferences with radii of tolerance of 0.42% in the actual 4 units possible building.74 even cases one Still, in where proposals hold up to such scrutiny, consideration deserves recognition. There is, of course, a gap between our method of verifying the plan through analytic geometry and the ancient method of converting the location of its features into magnitudes for case at as m a 74. In the of the 3-unit radii circumference [13.707,23.279], of 0.081 in the y dimension is tol m erance (Fig. 19), it has already been established given by the distance of 4.636 from of 0.7%. naos corners are set m to case that the 9.062 the midpoint of the plan the external In the of the 5-unit radii the cross-axis at naos 12.101 m ra from [9.071,12.216]. wall of the and the (Fig. 21), the intersection of the central a m western Because the plan is symmetrical, only dius, resulting in distance of 11.178 circle and the exterior anta one corner naos to be con the baseline to the intersection in corner at of the needs from is [13.749,26.483], resulting to dimension = 4.6362 + in a m sidered here: from [0,12.216] the they (12.1012 y2). radius of 15.124 from [9.071, naos corner at [4.435,4.430] we find x The western wall's intersection with the 12.101] (see n. 60, above) and a dis dimensions of 4.435 and 7.786 m lateral circumference occurs at m andjy [13.707, tance of 14.382 from [13.749, to a as a calculate diagonal distance of 23.360], given by the wall's distance 12.101]. If the circle with radius of 8.961 a of 1.1% m outer m at m, showing difference of 4.435 from the edge of the 15.124 is centered the end of the from the 9.062 m. and the 12.101 m x-x at expected euthynteria radius, theoretical baseline [18.142, case 4-unit in a m In the of the radii resulting distance of 11.259 from 12.101], itwill intersectwith the line of we to intersection in anta at a (Fig. 20), may reference the line of the baseline the the the [13.749,26.573], differ the = + ence m either long pronaos wall. That of y dimension (12.1012 4.4352 y2). of 0.09 from [13.749,26.483], western wall intersects with the central The difference of these intersections or an error of 0.6%. 584 JOHN R. SENSENEY -- I I r I I -r 10om an 21. Restored theoretical the actual building. We might, therefore, ask how architectural scale Figure plan of A overlaid with intersec drawing would have been created in the Hellenistic period. This question Temple tions of circumferences with radii of is relevant to the of Doric temples, where interstitial especially planning 5 units columnar contraction precluded convenient repetition of uniform paving ensure a slabs that conformity to grid-based plan. answer case a A reasonable in the of Temple A, I suggest, lies in simple intuitive process that begins with the initial schematic sketch before the completion of the detailed drawing (see Fig. 22): (1) within the smaller circle, set the lines of the exterior walls of the naos at the rear and sides to with approximately equal distances the outside edges of the overall plan in accordance with the principle of symmetry; (2) where the lateral lines same again intersect with the circumference of this circle, set the spur walls naos same separating the and pronaos; (3) in conjunction with these lateral lines, set the antae at the intersection with the circumference of the larger circle. In the drawing process itself, this result is most easily achieved in away that is similar to what I describe above: first set the locations of the corners s and the antae by establishing equal distances from the plan edges, on and then mark these points with the compass set the termini and center x-x. of the baseline In these ways, the logic of the overall design maintains are symmetry with interrelationships that circumferential, which is in keep a ing with process of drawing that relies upon the rule and compass. IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 585 - - - I- - I - - \ I --1 Figure 22. Proposed geometric underpinning of Temple A we a If again consider the hypothesis of modular-based metrology, we can on one manner s speculate in which the. plan designer might have established scale. Since the placement of features depends upon circum as ferential considerations, while the production of elements such paving slabs must be related to orthogonal dimensions, itwould appear that the a drawing precedes the scaling in the following way. In privileging fixed as axes magnitude such 45 Doric feet for the colonnade of the front and measure rear, the architect could the remaining elements in the drawing as (such the dimensions and placements of the walls and the varying dimensions of the individual slabs that make up the stylobate and steps) against these established distances and fix their sizes according to scale. By its nature, this procedure would be inexact for two reasons. In the first place, the expectation of symmetry in the final built form would dictate naos to equal values for the distances from the exterior walls the edge of the stylobate at both the sides and rear,when in fact the geometry of the drawn a rear form would show very slight discrepancy between the lateral and distances; indeed, the separation of 0.38% in the centers of the theoretical to a circumferences (Fig. 18) is likely be result of this very consideration.75 75. Two in which the archi measure on ways Secondly, the plans designer would need to the features the tect could resolve this issue would be drawing surface by hand and convert them to varying values. Unlike the to center the for the (1) compass case with Ionic the varied of columns in a Doric smaller circumference at a dif temples, spacing temple slightly such as that at Kos dictated that individual slabs could not an ferent location (see discussion above), repeat or to a established would need to be subdivided (2) verbally designate larger prototype. Distances, therefore, numerical distance for the area behind into varying units for the paving slabs in accordance with the spatial the naos, and subtract this distance contractions. from the of the walls of the length In the end, therefore, the measurements would have needed to ad The latter solution seems both pronaos. dress the individual slabs in addition to the overall size of the more and more paving practical probable, or Because of the in this especially considering Greek traditions stylobate euthynteria. multiple steps process, of verbal in and the modifications bound to occur in each of these it is specification "incomplete slight steps, as preliminary planning," discussed by not reasonable for us to theorize intended values for each element and Coulton (1985). In either case the ad as measurements dimension of the plan, given down to the dactyl. Instead, is both in relation justment very small, the result of this remains the revealed to the tolerance of 0.42% in significant study correspondence expected of the overall form to a theoretical in which the the final built form and in the theoreti rational, geometry per cal distance it would to in of error remain within the strictest tolerance found in correspond centages possible the scale the actual construction. original drawing. 586 JOHN R. SENSENEY CONCLUSIONS A metrological analysis of Temple A in the Asklepieion at Kos suggests that the triglyph module theory proposed byWilson Jones for 5th-century Doric temples may be applicable to this Hellenistic example. This theory cannot, however, account for the locations of features not associated with as naos the temples trabeation, such the walls of the and pronaos. Since was in an era Temple A created when the kind of drawn plan described by Vitruvius is likely to have been already commonplace in Ionic temples, we are justified in asking how its plan might address the considerations of design particular to the Doric order, where transparent orthogonal relation a were not ships established with grid possible. A geometric analysis that to responds the methodological issues addressed by Korres demonstrates a that circumscribed Pythagorean triangle forms the basis of Temple As design, in which circumferences determine the placements of the plans more principal features. Unlike the difficult problem of verifying the rests modular theory, the evidence for this geometric system solely upon to norms. the internal, measurable correspondences that conform Euclidian we can Furthermore, replicate these results both by calculation and with we can CAD software. In combination with the modular theory, speculate axes a that the colonnade may have played role in establishing scale in the drawn plan, by allowing for the conversion of relative dimensions into actual values for the building. The full implications of the results of this a analysis cannot be explored in the present study, but few observations merit brief comment.76 runs counter to In its details, the design process proposed here the as as cur simple grid approach used in Ionic temples, well differing from were rent ideas about the way in which Doric temples designed. Wilson on Jones insists the principle of "facade-driven" design for Doric temples, in contrast to the "plan-driven" design for Ionic temples.77 In other words, to architects designed Doric temples strictly according the commensuration as a of elements in the facade, opposed to the creation of guiding plan that determined the layouts of Ionic temples. Yet given the mixing of the archi tectural orders as as the 5th b.c.?most witnessed early century famously in the Parthenon?we such notions of mutual an might question categorical 76. In article currently in prog in as late as the Hellenistic As exclusiveness, particularly buildings period. ress, I assess the results of the present discussed above, it appears that the triglyph module may very well have analysis along with other considerations a in the context of ancient Greek played significant role in the design of Temple As facade. One might larger architectural masonry tools, wonder, however, ancient architects who are to have been trained drawing, why likely and methods of in the details of both orders should have planning. necessarily repressed planning 77.Wilson Jones 2000b, pp. 64-65; to a tendencies solely due the employment of particular module. After all, 2001. notion of mutu as Ingrid Rowland has convincingly demonstrated the very 78. The notion of the "orders" to an defined to ally exclusive "orders" be early modern transformation of Vitruvius's rigidly categories appears with Renaissance thinkers in genera, which, like ancient buildings themselves, accommodate notable begin the milieu of and omit a discussion Raphael Bramante, degrees of interchangeability.78 That Vitruvius should continuing later with Serlio, Palladio, of taxis in relation to Doric reflects his bias toward the temples probably and Vignola; see Rowland 1994; Howe traditions of Ionic that formed the core of his architectural train design and Rowland 1999, p. 15. s taxis from limited 79. See Tomlinson 1963. ing.79 If Vitruvius ignorance of Doric stemmed this IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 587 no reason we background, there is why should perpetuate his ignorance by extending it retrospectively to Hellenistic architects and their buildings. With support from the results of this analysis, it is even worth specu on a lating the special potential of the Doric order for higher degree of case sophistication in the drawing-board design process. At least in the of a Temple A, the variations in columnar placements in Doric temple might an have motivated alternative approach to the location of the internal to was a features within the plan.80 What appears have resulted system more interesting than the simple arithmetical relationships characteristic one was too reuse of the grid plan, but also that perhaps innovative for and continued development. Perhaps partly for this reason, and partly because of the "decline" in the production of Doric temples altogether, the possibly Doric-related method found in Temple A may have disappeared from common practice well before Vitruvius picked up his pen. Yet Temple A was not the final instance of this approach, which appears to have extended even a beyond the Doric order and into Hellenistic-Roman context, where to temple plans continued demonstrate the application of the Pythagorean as triangle and 3:5 circumferences their guiding geometry.81 Ultimately, however, the geometry of form characteristic of Temple A might have its not most recognizable legacy in the cryptomethodic taxis of the architect's drawing board, but in the shapes that Roman opus caementicium finally al lowed for permanent expression in three dimensions. Framed in this way, the fully experiential intersection of the idea and its reflection would give to a new rise aesthetic that would have been unimaginable in the Hellenistic architectural theory of Vitruvius. 80. In the lateral corner 81. are Temple A, These geometric approaches Coarelli 1987, pp. 11-21. For the round interaxes measure ca. in two the see 1973. column 2.7 m, found of earliest hellenizing temple, Rakob and Heilmeyer as to the other inter opposed average temples in Italy during the Republican An elaborated analysis of such geom axes of ca. 3.05 m. In the facade and at its the connec period: theTemple of Juno Gabii of etry, significance, and rear corner ca. b.c. colonnades, the column 160 and the round temple of tions between these examples and the measure ca. ca. interaxes 2.7 m, while 120-100 b.c. in Rome's Forum Boa work at Kos discussed in the present the second and central interaxes mea rium. For of the are I in a the geometry Temple study themes that explore sure 3.065 and 3.080 at see m, respectively; of Juno Gabii, Almagro-Gorbea follow-up article (in progress) focused see on Schazmann and Herzog 1932, 1982; Jim?nez 1982, esp. pp. 63-74; Roman architecture. 2. pi. Almagro-Gorbea and Jim?nez 1982; APPENDIX 1 THEORETICAL PLAN A Select locations, coordinates, magnitudes, and equations for theoretical are to plan A given below. The coordinates correspond measurements in meters taken by Schazmann and Herzog (see Fig. 5),82 converted here to an x 18.142 33.280 quadrant with origin 0, 0 and limit 18.142, 33.280 at the southeastern and northwestern extremes, respectively (Fig. 23). For see additional equations, text and notes above. Location Relevant Circumference Coordinates 1 A 0,0 1 B 18.142,0 1 C 24.202 D 2 4.435,4.430 E 2 13.707,4.430 F 2 13.707,20.002 G 2 4.435,20.002 H 1 13.749,26.483 I 1 4.393,26.483 1 Circumference (Circumcenter 9.071,12.101) Definitions AC = Diameter = Radius 1 + Radius 2 = Radius 1 distance from circumcenter to A (or B) = Radius 2 distance from circumcenter to H (or I) Magnitudes X, Y distances from circumcenter to A XI: 9.071-0 = 9.071 Yl: 12.101-0 = 12.101 X, Y distances from circumcenter to H - X2:13.749 9.071 = 4.678 82. Schazmann and Herzog 1932, = Y2:26.483-12.101 14.382 pi. 2. IN IDEA AND VISUALITY HELLENISTIC ARCHITECTURE 589 / ^ID EIL ,-_ G F_ _ - __ 0 _~ 1^L _ O O- I 0 Figure 23. Restored theoretical plan of Temple A with geometric under and indicated locations cor pinning 0WJ~ ~10mlomo to Cartesian coordinates responding in Equations Differences Magnitudes = Radiusl2 = Xl2 +Yl2 30.247-30.205 0.042 = Radius 12=9.0712+ 12.1012 18.148-18.123 0.025 Radius 1 = 15.123 Tolerances = + Radius 22 X22 Y22 0.042 / 30.247 = 0.1% Radius 22=4.6782+14.3822 0.025 / 18.123 = 0.1% Radius 2 = 15.124 AC = 15.123 + 15.124 = 30.247 Pythagorean Triangle Equations = + x 2 hypotenuse2 AB2 (Yl 2)2 Circumference = 18.1422 + 24.2022 (Circumcenter 9.071,12.216) hypotenuse2 = hypotenuse 30.247 Diameter2 = (DE)2 + (EF)2 = Diameter2 9.2722+ 15.5722 Difference inMagnitude = - Diameter 18.123 AC hypotenuse - 30.247 30.247 = 0 6:10 Ratio of Circumferences 1 and 2 Equations Tolerance = = (18.123 / 6) x 10 30.205 0 / 30.247 0% = (30.247 / 10) x 6 18.148 APPENDIX 2 THEORETICAL PLAN B Select locations, coordinates, magnitudes, and equations for theoretical are plan B given below. The coordinates correspond to measurements in an meters taken by Schazmann and Herzog (see Fig. 5),83 converted here to x 18.075 33.280 quadrant with origin 0,0 and limit 18.075,33.280 at the a southeastern and northwestern extremes, respectively, and symmetrically naos centered (see Fig. 23, scaled for the slightly differing dimensions and coordinates of Appendix 1). Location Relevant Circumference Coordinates 1 A 0,0 1 B 18.075,0 C 1 18.075,24.224 D 2 4.402,4.430 E 2 13.674,4.430 F 2 13.674,20.002 G 2 4.402,20.002 H 1 13.716,26.483 I 1 4.360,26.483 Circumference 1 (Circumcenter 9.038,12.112) Definitions AC = Diameter = Radius 1 + Radius 2 = Radius 1 distance from circumcenter to A (or B) = Radius 2 distance from circumcenter to H (or I) Magnitudes X, Y distances from circumcenter to A - XI: 9.038 0 = 9.038 Yl: 12.112-0 = 12.112 X, Y distances from circumcenter to H - = X2:13.716 9.038 4.678 83. Schazmann and 1932, - Herzog = 2. Y2: 26.483 12.112 14.371 pi. IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 59I Equations Differences inMagnitudes - Radiusl2 =Xl2 +Yl2 30.225 30.205 = 0.020 Radius 12=9.0382+12.1122 18.135 -18.123 = 0.012 Radius 1 = 15.112 Tolerances Radius 22 =X22 +Y22 0.020 / 30.225 < 0.1% Radius 22=4.6782+14.3712 0.012 / 18.123 < 0.1% Radius 2 = 15.113 AC = 15.112 + 15.113 = 30.225 Pythagorean Triangle Equations = 2 AB2 + (Yl x 2)2 Circumference hypotenuse2 = 18.0752+ 24.2242 (Circumcenter 9.038,12.216) hypotenuse2 = 30.224 = hypotenuse Diameter2 (DE)2 + (EF)2 Diameter2 = 9.2722+ 15.5722 Difference inMagnitude Diameter = 18.123 - AC hypotenuse - 30.225 30.224 = 0.001 1 6:10 Ratio of Circumferences and 2 Equations Tolerance = (18.123 / 6) x 10 30.205 0.001 / 30.224 < 0.1% (30.225 / 10) x 6 = 18.135 APPENDIX 3 THEORETICAL PLAN C Select locations, coordinates, magnitudes, and equations for theoretical are plan C given below. The coordinates correspond to measurements in meters taken by Schazmann and Herzog (see Fig. 5),84 converted here to an x 18.075 33.280 quadrant with origin 0, 0 and limit 18.075, 33.280 at the southeastern and northwestern extremes, respectively, with the width of naos m the reduced .067 in order to provide symmetry (see Fig. 23, scaled for the slightly differing dimensions and coordinates of Appendix 1). Location Relevant Circumference Coordinates 1 A 0,0 1 B 18.075,0 C 1 18.075,24.224 D 2 4.435,4.430 E 2 13.640,4.430 F 2 13.640,20.002 G 2 4.435,20.002 H 1 13.682,26.483 I 1 4.393,26.483 Circumference 1 (Circumcenter 9.038,12.107) Definitions AC = Diameter = Radius 1 + Radius 2 = Radius 1 distance from circumcenter to A (or B) = Radius 2 distance from circumcenter to H (or I) Magnitudes X, Y distances from circumcenter to A XI: 9.038-0 = 9.038 Yl: 12.107-0 = 12.107 X, Y distances from circumcenter to H - 9.038 = 4.644 X2:13.682 84. Schazmann and Herzog 1932, - = Y2: 26.483 12.107 14.376 pi. 2. IDEA AND VISUALITY IN HELLENISTIC ARCHITECTURE 593 Equations Differences inMagnitudes - Radiusl2=Xl2 +Yl2 30.216 30.148 = 0.068 Radius 12=9.0382+12.1072 18.130-18.089 = 0.041 Radius 1 = 15.108 Tolerances Radius 22 =X22 +Y22 0.068 / 30.216 = 0.2% Radius 22=4.6442+14.3762 0.041 / 18.089 = 0.2% Radius 2 = 15.108 AC = 15.108 + 15.108 = 30.216 Pythagorean Triangle Equations = 2 AB2 + x Circumference hypotenuse2 (Yl 2)2 = 18.0752+ 24.2142 (Circumcenter 9.038,12.216) hypotenuse2 = 30.216 = hypotenuse Diameter2 (DE)2 + (EF)2 Diameter2 = 9.2052+ 15.5722 Difference inMagnitude Diameter = 18.089 - AC hypotenuse - 30.216 30.216 = 0 6:10 Ratio of Circumferences 1 and 2 Equations Tolerance x = (18.089 / 6) 10 30.148 0/30.216 = 0% (30.216 / 10) x 6 = 18.130 594 JOHN R. SENSENEY REFERENCES 1982. 20-23 1987 Almagro-Gorbea, M. "Levanta ture, Leiden, January miento ed. H. 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(Hypomnemata 51), G?ttingen. R. John Senseney University of Illinois at Urbana-Champaign school of architecture temple 117 hoyne buell hall 611 taft drive, mc 621 champaign, illinois 6182o