Leon Battista Alberti and the Homogeneity of Space Author(s): Branko Mitrović Source: Journal of the Society of Architectural Historians, Vol. 63, No. 4 (Dec., 2004), pp. 424- 439 Published by: University of California Press on behalf of the Society of Architectural Historians Stable URL: http://www.jstor.org/stable/4128013 . Accessed: 10/02/2014 11:35

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This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions Leon Battista Alberti and the Homogeneityof Space

BRANKO MITROVIC Unitec Instituteof Technology

He thought, with most people, that everything is somewhere the historyof the visualarts, the processwhereby the under- and in place. If this is its nature, the power of place must be a standing of space as homogenous came about. He believed marvelous thing, and be priorto all other things. For that with- that the conception of space as homogenous and systematic out which nothing else can exist, while it can exist without the arose shortly before the discoveryof the geometricalcon- others, must needs be first; for place does not pass out of exis- struction of perspective.4In later years, a position similar tence when the things in it are annihilated. to Panofsky'shas been defended by SamuelEdgerton, who, Aristotle, Physics' in his RenaissanceRediscovery ofLinear Perspective,argued that a "'systematicspace' infinite, homogenous and isotropic," made possible "theadvent of linearperspective."' However, The homogeneityof spacewas firstdiscussed as a a body of more recent scholarshiphas denied this view and philosophicalproblem by Ernst Cassirer,and the claimedthat the understandingof spaceas homogenouswas related theoretical considerations were subse- a post-Renaissancedevelopment. The debate has complex quently introduced into architectural and art history by implicationsnot only for the history of perspectivebut also Erwin Panofsky in "Perspective as Symbolic Form."2 for the understanding of Renaissance architecture and Panofskyassumed that in order to construct geometrically architecturaltheory. Had Renaissancearchitects and theo- a perspectivaldrawing, one must postulate space as a con- rists indeed conceived of space as heterogeneous,' they sistent medium in which the depicted objects are located. could not have believed that the same shapes (say, of the The definitionof homogenous space that Panofskyadopted classicalorders) were reproduciblein differentlocations. If from Cassirer had two parts.3The first section stipulated one assumesthe heterogeneityof space,it is very difficultto that all elements of a space-points and sets of points-are operate with the concept of shape as it is normally under- mere designations of positions. They do not possess any stood. In a heterogeneous space, there would exist points other content except their position relative to each other on a shape whose distancescould not be quantifiedor geo- and their existence is not substantialbut purely functional. metrically comparedto distances between other points on The second partof Cassirer'sdefinition formulated the pos- the same shape. There would be no possibility of making tulate of homogeneity,which states that from every point in the same shapesat differentlocations, nor could one repro- space it must be possible to draw identical figures. duce the same shape by replicatingits geometrical disposi- Panofsky'sefforts in "Perspectiveas Symbolic Form" tion of lines, angles, and surfaces.If Renaissancearchitects were directed toward establishingand describing,through and architecturaltheorists indeed believed in the hetero-

This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions geneity of space, and consequently did not have the con- until after the Renaissance."o1 He also cited Peter Collins's cept of shape as it is normally understood, then it becomes observation: "It is a curious fact that until the eighteenth extremely difficult to explain their efforts to define sizes and century no architectural treatise ever used the word geometrical relationships between the elements of the clas- 'space."'"1 Methodologically speaking, it would not be sical orders in order to reproduce them.' One aspect of incorrect to dismiss Elkins's and Collins's positions because these efforts, for instance in the case of Palladio and Vig- they confuse the concepts used in the analysis with the nola, was the development of a system of presentation of assumptions these concepts are meant to analyze. Elkins architectural elements that combined plans, sections, and admits that Panofsky's concepts describe a set of assump- elevations in one drawing. The drawings in Figures 1 and 2 tions that can be observed in Renaissance paintings.12 Say- cannot be understood if one assumes that they represent ing that such concepts cannot be used retrospectively is like shapes in a heterogeneous space. arguing that one cannot say "Columbus discovered the The idea that Renaissance architects and architectural American continent" because at the time of the discovery, theorists assumed the heterogeneity of space and did not the concept "American continent" was unknown. Peter therefore have the concept of shape ultimately means that Collins's argument is even weaker: because Renaissance the- the shapes of architectural elements-the formal and visual orists did not use the word "space," they could not conceive properties of architectural works-are irrelevant in the of space-the claim is not that the word was used differ- study of Renaissance architecture. It would follow that it is ently than it is today, but that the lack of its use indicates the pointless for architectural history to study these properties absence of the corresponding idea. For this argument to be in Renaissance buildings and that the discipline must be valid, one must assume that people cannot have certain reduced to the reconstruction of the verbal behavior that ideas if they do not name them the same way as we do. architectural works prompted at the time they were built- Methodological problems of this kind are abundant in that one can study only the narratives or "meanings" associ- the debate about the history of understanding space as ated with buildings." homogenous. They often result from the fact that the impli- The question of whether Leon Battista Alberti, in his cations of homogeneity are commonsensical, easily taken treatises on painting, sculpture, and architecture, was able to for granted and overlooked. It is not enough to say that dur- conceive of three-dimensional, homogenous space is cru- ing the Renaissance, space was understood as heteroge- cial for the outcome of this debate.9 Alberti was the first to neous: one has to explain how Renaissance theorists and provide a written description of the geometrical construc- artists could have believed that the geometrical description tion of perspective, and if one could show that his views of visual and spatial experience was possible if they did not relied on the assumption of the homogeneity of space, then believe that the totality of spatial relationships between the program that reduces the study of Renaissance archi- shapes could be geometrically defined. This applies not only tecture exclusively to the study of narratives attached to to perspective. The complex systems of coordinated plans, architectural works would be unjustified. Conversely, if he sections, and elevations, such as those developed in Palla- did not have the concept of homogenous space, it should dio's and Vignola's architectural treatises, relied on the be immensely interesting to see not only how he managed assumption that the totality of a shape could be defined by to formulate and justify the use of geometry in the con- mathematical determination of all relationships between its struction of perspective, but also how he conceived of archi- lines and angles-and also that readers would interpret the tecture and architectural theory in a heterogeneous space. drawings of the classical orders starting from that assump- tion. Palladio's drawing of the details of the Ionic order (see 1) the between Debate about the of Some Figure carefully exploits homology"3 plan, Homogeneity Space: section, and elevation. Elements of ornamentation are not Considerations Methodological merely shown from different sides; different projections are Contrary to the view of scholars such as Panofsky and carefully coordinated so that, for example, the position of Edgerton, a number of more recent authors have claimed one edge of the abacus in plan corresponds to its position in that during the Renaissancespace was not conceived of as elevation, whereas another edge, which is a line in plan, homogenous. James Elkins, for instance, has argued that appears only as a point in elevation. The width of flutings, the understandingof space as homogenous developed long presented in full size in plan, appears shortened in eleva- after the Renaissance and noted that the concepts of Panof- tion, exactly the way rules for orthogonal projection would sky's analysis ("systematic" or "homogenous" space, and so require. All this enables the drawing to be read as a com- forth) "are all modern and do not occur in mathematics plete and consistent description of a given shape-some-

LEON BATTISTA ALBERTI AND THE HOMOGENEITY OF SPACE 425

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thing that would not be possible if the heterogeneity of tions of the claim that the architects of the past were not space were assumed. awareof the homogeneityof space.The authors'wider claim Similardifficulties follow if one ascribesthe belief in the is that "the hypothesisof a homogenous space, with its sys- heterogeneityof space to quattrocentoarchitects and theo- tem of spatial coordinates among plan, section and eleva- rists. In their ArchitecturalRepresentation and the Perspective tion, did not appearuntil the eighteenth century."15is"In the Hinge,Alberto P6rez-G6mez and Louise Pelletier state that fifteenth century,the growing fascinationof painterswith "Brunelleschi'sexperience shows that he could not conceive linearperspective did not lead to a geometricsystematization of a building in a homogenous space."14Perez-G6mez and of pictorialdepth, nor did it instrumentalizethe process of Pelletier'sbook makes a particularlyvaluable contribution creation.The world of everydayexperience relied on quali- to the debate becauseit exploresthe most extremeimplica- tatively distinct places and poetic narrativesthat integrated

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the golden age of antiquity with the current cosmological ern scientific worldview nor could it have been there to assist order. Homogenous space could exist only in the supralu- the discovery of the geometrical construction of perspective. nar realm, where the movements of the heavenly bodies pro- People lived (and architects designed) happily without vided a normative order for auspicious action in the human knowing that they inhabited a homogenous space, or, as we realm of constant change and corruption."l16 The ultimate are left to infer, the idea ("hypothesis," according to the implication of the argument is that the idea of space as we authors) that we inhabit a homogenous space is a cultural know it today came about a couple of centuries after the construct and an unfortunate byproduct of modern positivist Renaissance and is merely a product of modern science and and scientific Weltanschauung. its efforts to provide a rational and mathematical description However, there can be no knowledge of-let alone "fas- of the world. The idea did not precede the rise of the mod- cination" with-linear perspective without "geometric sys-

LEON BATTISTA ALBERTI AND THE HOMOGENEITY OF SPACE 427

This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions tematization of pictorial depth." It is unclear how architects tance from A to B in that case would not be the same as from could have designed if they assumed that their buildings were B to A. So here is the rub: if one ascribes this view to in a heterogeneous space. For all authors who claimed that Brunelleschi, one has to explain how an architect who the concept of homogenous space was not available in the believed that the distance from one point to another could Renaissance, Brunelleschi's geometrical construction of per- be different from the distance from the latter to the former spective has traditionally been very difficult to explain. If could have conceived of and designed the geometry of the Brunelleschi did not discover the geometrical construction dome of the Cathedral of Florence. Would Brunelleschi be of perspective, then one has to rewrite much of Renaissance incapable of calculating the quantities of the material needed art history-but if he did, then it is difficult to say how he did to build the dome? The number of difficulties one can imag- it by assuming the heterogeneity of space." Brunelleschi's ine is legion, and, with all due respect to cultural construc- perspectival procedures are not the topic of this article, but tivism, if the claim that Brunelleschi "could not conceive of the argument that P6rez-G6mez and Pelletier use to dismiss a building in a homogenous space" is to be credible, one has Vasari's report that Brunelleschi geometrically constructed to explain the paradoxes first. his drawing starting from the plan points to the heart of the These difficulties are only the tip of the iceberg. The problem. The authors assert: "From the point of architec- problem is Panofsky's as well. Panofsky repeated Cassirer's tural design ... the potential homology among plan, eleva- definition of homogenous space, including the homogene- tion and perspective as forms of visual projection was not ity postulate. Insofar as he believed that the understanding immediately realized."'1 We shall see later that Alberti actu- of space as homogenous immediately preceded the discov- ally stated that his procedure was to initiate the drawings by ery of the geometrical construction of perspective, he inscribing the building's plan in the perspective19-but the should have explained how medieval and ancient architects real problem is in the wider implications of the authors' built and designed while assuming that a building could claim. Their position implies, for instance, that Brunelleschi have different lengths if measured from one side rather was not aware that one and the same building could not have from another. P6rez-G6mez and Pelletier have thus merely material different faqade lengths when represented in plan and eleva- extended to the Renaissance and post-Renaissance tion, or that he would not think that something was wrong if an interpretive problem that is inherent in Panofsky's a plan were to show a building with one door, whereas two account as well. doors would appear in the elevation.20 This interpretation of Panofsky overlooked the fact that Cassirer's definition P rez-G6mez and Pelletier's argument, however farfetched, was actually bipartite. Saying that architects or theorists of is not a misunderstanding-it directly follows from the the past did not have the concept of homogenous space can authors' view that "the descriptive sets of projections that we mean that they disagreed with the first part of the defini- take for granted operate in a geometrized, homogenous space tion (the idea of space) or the second (the homogeneity pos- that was construed as the 'real' space of human action during tulate) or both. Since Panofsky was writing about the impact the nineteenth century."21 of the understanding of space on perspective, it was the sec- Their view may seem paradoxical, but in fact Perez- ond part that really mattered for his argument. One needs G6mez and Pelletier are to be credited with having consis- homogeneity, not necessarily the concept of space, for the tently developed a position whose problems were already geometrical construction of a perspectival drawing. Panof- implicitly present in Cassirer's definition. Such a view indeed sky took it for granted, however, that without space there follows if one adopts Cassirer's definition of the homogene- could be no homogeneity. When he reviewed ancient space ity of space and then, contrary to Panofsky, claims that this theories, he was satisfied simply to show that the philoso- conception of space (with its implications of the homology phers he considered did not assume the continuity (that is, among plan, section, and elevation) was not available (its homogeneity) ofspace. He did not take into account that the implications not yet "realized") in the Renaissance. It will be ancients could have endorsed all the geometrical implica- remembered that the second part of Cassirer's definition (the tions of the homogeneity of space without actually relying homogeneity postulate) stipulated that in a homogenous on the concept of space.22 At first, Panofsky's reasoning space it must be possible to draw the same figures from every looks plausible: the idea that one could somehow conceive point in space. A "figure" can be a simple line. In a space in of homogeneity without space is counterintuitive. Homo- which it was not possible to draw identical figures from every geneity is a relationship between dimensions, dimensions point and in every direction, there would exist at least two are spatial relationships, therefore there can be no homo- points A and B such that one could draw a straight line of a geneity without space. (If space did not exist, how could it definite length from A to B, but not from B to A. The dis- be homogenous?) However, even if we accept the claim that

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This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions some people in the past did not have the concept of space, each thing.26Plato-whom Aristotlecredits with havingbeen it is still plausible that they realized that a thing has the same the first to ask the question-suggested that place is matter, length from whatever side it is measured and that the same on which the form is imprinted.27An alternative answer figures can be drawn (or shapes carved) wherever one is would be that it is the form of an object. But Aristotle dis- located. The absence of a theoretical concept cannot be missed both views becauseplace must be separablefrom the taken for the unawareness of the phenomena the concept object,whereas form and matterare not; also,if form or mat- was subsequently constructed to explain. ter were its place, a thing could not be said to move into its To put all of this in simple words: things have dimen- place.28Both the form and place of a body are its limits, but sions and there are certain rules about how dimensions can the form belongs to the body itself,while its placeis the limit be compared consistently. Cassirer and Panofsky have called of the body that surroundsit.29 Aristotle also dismissedthe these rules "the homogeneity of space." Their "postulate" is possibilitythat the place of a body can be a dimensionor the a sentence from which all (or many) of these rules can be extension between its external limits-a view that clearly deduced. But this does not mean that people who operated relates to our modern idea of "space."(The word Aristotle without the concept of space did not know how to measure used here, diastema,can be translatedas "interval"or "exten- things, that they were not aware of the manifestationsof the sion";in Euclid it came to mean "radius."30)The argument "homogeneity of space." Similarly, philosophers who did not Aristotle advancedagainst this view shows the depth of the rely on the concept of space could have formulated a theory rift between what he meant by "place"and the understand- about dimensions that would endorse and account for all the ing of place as a part of space. This idea is wrong, Aristotle commonly known manifestations of the homogeneity of argued,because if a purely dimensionalentity, an extension space without relying on the concept of space. One could that does not belong to a body, could exist, it would follow call such a position "homogeneity without space." What that partsof a fluid moving in a containerwould producean Panofsky overlooked was that this position was advocated by infinite number of overlappingplaces: in other words, every the greatest authority in matters of philosophy the Middle particleof fluid would leave a place-that is, an immaterial Ages and Renaissance knew of: Aristotle himself. dimensional entity-behind itself after moving to another place.This argumentis incomprehensibleif one forgets that for Aristotle a dimension (extension)is alwaysa dimension Aristotle on and Space, Place, Homogeneity (extension)of something:it is a quantityand quantitiescan- Insofar as Renaissance authors were exposed to the view not exist independently of substances.Even if a substance that objects in the world are not in a homogenous space,this could leaveits dimensionwhen moving to anotherplace, this idea came from Aristotle and had a specific role within the dimension would still belong to its original substance- Aristotelian system. Aristotle's Physics operated with places, otherwisethis dimensionwould exist as an immaterialentity, but did not assume the existence of space.The first five chap- something Aristotle'scritique of Plato's immaterialForms ters of the fourth book of the Physicsexplain that the world precluded.In Aristotle'sview, place thus has to be the inner- consists of places into which bodies move and which are all most surface(the limit or the extremeboundary) of the body. contained in the totality of the world.23 Some places are There cannot exist such a thing as an immaterialextension above, others below; places even have power (dynamis)with between parts of the surface that would not belong to the those above attracting what is light, those below what is body contained.31 heavy.24 Aristotle also argued that it is wrong to say that Aristotle had important concerns pertaining to the when a container of water moves from one place to another, structureof his systemwhen he developedhis teachingabout the water in it changes place; rather, it is more proper to say places. Saying that an immaterialentity such as space exists that the water remains in the same place (or, in the con- comes dangerouslyclose to saying that nothing exists.32If tainer). WNhenwater replaces air in a container, we refer to he allowed the existence of such an entity as space without the place in which air was and in which water is now. We do matter,it would follow that void, vacuum,or a placewithout not refer to a place as ultimately determined by its rela- matteris possible. The possibilityof void, a place without a tionship to the whole cosmic place-that is, we should not body, is discussedand dismissedin the Physicsimmediately conceive of places as units of space.25 Place, for Aristotle, is after the explanation of what is place.33Aristotle's central not a dimensional fragment of space merely defined by its argumentis that if void could exist, then objects in it would geometrical relationship to other places. move with an infinite speed: the speed of an object is pro- According to Aristotle, there are four possible answers to portionally inverse to the resistance of the medium and if the question of what place is, what the vessel is that contains there is no resistancethe speed has to be infinite.34

LEON BATTISTA ALBERTI AND THE HOMOGENEITY OF SPACE 429

This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions But there were even greater problems with the concept the problemwas accountingfor homogeneity without rely- of space that pertained to the very structure of the Aris- ing on immaterialentities such as space. When Aristotle totelian system. It would be quite wrong to think that Aris- denied the existenceof space,his point was that one does not totle could not conceive of space as an immaterial medium need to postulateit in order to explainhow things can have existing independently of material objects. The fragment dimensions(diastemata). He did not deny the manifestations from the Physicscited as the motto of this paper clearly for- of what Cassirerand Panofskyhave called"the homogeneity mulated this view, ascribing it to Hesiod and "most people."35 of space."In his view,all thingsare in placesand theirdimen- Aristotle certainly did know about the idea, but his account sions belong to them as bodies. He certainlydid not suggest in the section cited sounds sardonic and he described the that one and the same dimensionof a body can be different position merely in order to dismiss it. In one of his less fair- depending on the side from which it is measured. The minded moods, Aristotle continued the section by arguing lengths of two bodies can be compared,he saysin the Cate- that it is unclear how such an entity could exist.36On the one gories,because it is particularto quantitiesbelonging to dif- hand, if it is immaterial, it is unclear how it can have dimen- ferent bodiesthat they can be comparedin the sense of equal sions-dimensions in his view belong to bodies and deter- and not-equal.41Aristotle said that all quantitiescan be com- mine their limits. On the other, if it is material, it must have paredas equaland not-equal.42Obviously, this would not be its own place and be in it at the same time with the material possible had he believed that identical quantitiescould be body it contains-in which case two bodies will share the equal or not-equaldepending on their location.The section same place. The section allows one to sense the numerous about the comparisonof quantitiesas equaland not-equalin ontological concerns which motivated Aristotle's position. He the Categoriescan thus be taken as endorsementof homo- dismissed Plato's immaterial Ideas and insofar as he tried to geneity and de facto it impliesthat from everypoint belong- avoid similarly immaterial entities in his metaphysics, he was ing to every place it is possible to draw identical figures.43 unlikely to welcome an immaterial three-dimensional Since dimensionsof lines and anglesare quantities,this same medium in which physical bodies are placed, and one that section from the Categoriesalone is enough to precludethe ensures (how? by what means?) that they behave according to argument that homology between orthogonal projections the rules of geometry. If a place (as a part of space) has dimen- (such as we have seen in Palladio'sdrawing) would be incon- sions (or extension) on its own, independently of the mater- ceivablein Renaissanceor pre-Renaissancetimes. Something ial bodies that fill it, then, since a dimension (or extension) is similar appliesto the geometricalconstruction of perspec- a quantity, it would follow that quantities can exist indepen- tive. Aristotle was mainly concerned with asserting that dently of substances. However, for Aristotle, quantity is a cat- dimensionscannot exist independently of the materialobject egory and must always belong to a material object (primary they belong to. Consequently,the distancebetween the eye substance)." This latter thesis was crucial for Aristotle in and the object perceived-and all other quantities,dimen- order to sustain his critique of Plato's theory of immaterial sions of lines, and angles necessaryfor the geometricalcon- Forms. The existence of space independent of the material structionofperspective-can be explainedas dimensionsthat objects it contains was thus incompatible with one of the belong to the airor some other transparentmedium, such as most fundamental strategies Aristotle adopted in his critique water. (Stuffs such as water or air can be conceived as of Plato's theory of Forms. This problem would have been homogenousinsofar they are infinitelydivisible into units of obvious to Aristotle's commentators. Among the ancients, equal density.)Aristotle was in any case explicit that vision Philoponus, whose views and influence on Renaissance can occur only in a materialmedium.44 As much as Panofsky thinkers I discuss below, made substantial efforts to combine was right in his realizationthat the geometricalconstruction the idea of homogenous space with the Aristotelian system.38 of perspectiverequired a homogenousmedium, he did over- In his Commentary on the Physics,he clearly formulated the look the possibilitythat this mediumneed not be immaterial problem mentioned but was merely able to say that "our space:it couldbe air.In Alberti'sFlorence, the latterview was agreed ideas" should be "consistent with the facts" and then clearly expressedby St. Antoninus, the prior of St. Marc's introduced the thesis that substances by themselves cannot conventfrom 1439 to 1444 andArchbishop of Florencefrom be taken to be self-hypostasized, but that they always require 1446 to 1459.4sIn his SummaTheologica, St. Antoninusdis- some determinate quantity for their being.39 His solution to cussed the necessaryrequirements of good vision and listed the problem thus sidestepped the concept of substance as the continuityof medium-which he identifiedwith air-as defined by Aristotle.40 one of them.46There is thus nothing in the teachingabout We have seen that Panofsky did not conceive of homo- places in Aristotle'sPhysics that would preventan accountof geneity without space-but for Aristotle, it could be said that the geometricalconstruction of perspective.As long as vision

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This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions occurs within the homogenous medium such as air, one can of space independent of the bodies it contains, or embraced describe it by mathematical models appropriate (or equiva- the Aristotelian position (homogeneity without space). It is lent) to those of homogenous space. A Renaissance theorist hard to imagine that he could have been unaware of the of perspective could have relied on the concept of homoge- manifestations of the principle Cassirer and Panofsky called nous space as Panofsky suggested-but need not have. An "the homogeneity of space." The belief that the same line Aristotelian explanation of the geometrical construction of will have different lengths if measured from one side rather perspective is equally possible. than another is unlikely to be found in the author who was None of this should be taken to mean that the Aris- the first to describe the geometrical construction of per- totelian system particularly contributed to the discovery of spective. Alberti's endorsement of the manifestations of the the geometrical construction of perspective. Aristotle's the- principle is particularly obvious if we take into account that ory of light, for instance, would have been of particularly his theory of perspective is part of his wider program of sys- little use. Light for Aristotle was the activity of the trans- tematic quantification of the topics he was writing about. parent qua transparent and not the result of a movement.47 Before we can answer the question about the understanding Such a position is unlikely to stimulate any study of visual of space (place) on which this program was based, it is nec- phenomena by means of a geometrical analysis of the lines essary to consider how the program worked. that connect the eye and points on the object perceived. It The account of perspective in Depictura is based on the was the long tradition of optical treatises-ancient, Arab, observation that the light rays that connect the eye with and medieval-that gave the necessary impetus for the dis- objects of perception travel in straight lines and that conse- covery of perspective. David Lindberg's seminal Theoriesof quently the perception of every line can be analyzed geo- Visionfrom Al-Kindi to Kepler provides a general survey of metrically, by means of a triangle whose base is the line this tradition.48 Lindberg has traced elements of Alberti's mentioned, and the opposing point the human eye.53 Every account of the visual phenomena (and especially the role of surface consists of lines, and the totality of our visual expe- the central ray) to the influence of Al-Hazan and the Bacon- rience is analyzable by means of geometry-that is, systems ian tradition.49 Similarly, Samuel Edgerton has related of triangles, two of whose points determine the ends of lines Alberti's method of the construction of the distance point to in space and the third of which is in our eye.s4 These trian- Euclid.50 Influences of the optical tradition can be thus gles constitute the pyramid of sight,5sswhich in turn con- traced in Alberti's De pictura even though Alberti bracketed sists of many smaller pyramids whose bases are individual the issue of the physical nature of sight.5s Lindberg noted in surfaces observed and whose apex is in the eye.56 The impli- his Theories of Vision: "Alberti's point is that the theory of cation is that all relationships between shapes of objects that linear perspective which he is about to develop, requires the we can experience visually can be described using geometry. visual pyramid, but need not concern itself with the direc- According to Alberti, the best of painters is the one who is tion of radiation or the functioning of the eye; it requires able to represent accurately proportions and differences mathematics, but not physics or physiology of vision"52- between surfaces.57 A picture is to be conceived as a section nevertheless, the very stimulus to use geometry for the pur- through the pyramid of sight-it is a plane that shows what pose was unlikely to come from the Aristotelian theory of we would see through a window located at the place of the vision. However, the topic of this article is not the way picture plane.ss8A perspectival drawing is the equivalent of Alberti derived his account of the geometrical construction the lines we would produce if we drew on the window the of perspective and the theories of vision which contributed outlines of the objects we see through it. The homology to it, but the theory of space which underlay it. At this stage, among plan and perspective is clearly assumed in Alberti's we can conclude that Aristotle's theory of places in itself description of the way he initiates the composition of a contains nothing inherently contradictory to the idea of the drawing by inscribing the plan of the foundations and walls geometrical construction of perspective. of the building in the pavement drawn in perspective.59 The necessary background assumption is that the geometrical construction of perspective describes the totality of visual- of the Medium Alberti and the Homogeneity spatial experience; we can anywhere assume that a window of Vision is placed between our eyes and the objects we observe-and Only after these preliminary considerations can one prop- consequently we can everywhere define, by means of the erly approach the question of Alberti's views on the homo- same geometrical transformations, the outlines of the geneity of space. Alberti could have fully subscribed to the objects on the picture plane as we would perceive them idea of the homogeneity of space and assumed the existence through the window glass. What we see will always be

LEON BATTISTA ALBERTI AND THE HOMOGENEITY OF SPACE 431

This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions definableby means of geometry.The nail is hit on the head communicationwas with words.Alberti was awareof all the in his Elementapicturae-a little essayintended, Alberti stip- difficultiesthat accompanyattempts to communicateabout ulated, to define combinationsof lines, angles, and surfaces visual and formalproperties in words;from time to time he so that there is nothing in nature that can be perceived by would include in his texts small drawings of architectural the eyes and yet not representedin lines.60 details that he could not describe verbally,or would even The programof the quantificationof visual experience describe certain shapes by referring to the shapes of let- coincides with Alberti'swider programof systematicquan- ters.65As Carpo puts it, Alberti'simportant problem was tification of the subjects he was writing about, developed how to convey by digital means the informationthat is bet- through his treatises on painting, sculpture, and architec- ter conveyed analogically.While visual and formalproper- ture. De statuadescribes three simple measuringdevices that ties playedan immenselyimportant role in his architectural Alberti invented to define geometricallythe totality of the theory, he had to find ways to describe these properties shapesof an object.Two of these devices serve to determine using numbersand letters and avoid analogousrepresenta- what Alberti called dimensioand the last onefinitio. Dimen- tion. Ultimately,this had to result in a programof quantifi- sio, Alberti explained,is the definite determinationof rela- cation of visual and spatial experience. His Descriptiourbis tionships between sizes, whereas finitio determines the Romaewas thus an exercise in defining the shape of Rome disposition of lines, angles, expanding and retracting sec- in numbers. The topic of De statuais the quantificationof tions of the body-that is, stipulatesthe externalboundaries the totality of spatialrelationships of a shape. And, to have and lines. Dimensiodetermines proportions of the parts of such a program,one has to believe that the quantificationof the body that tend to be equal for all individualsof a certain the totality of formal relationshipsis possible. It is hard to kind, while finitio pertains to those properties of the shape imagine that the programcould have been formulatedstart- that are characteristicfor an individual.6'1The former prac- ing from the assumptionthat it is not possible to drawthe tice suits those sculptors who want to make a representa- same figures from all points, or that in some cases the dis- tion of a human being in general,independently of whether tance between two points dependson the point from which it is Socratesor Plato; the latter to those artistswho want to it is measured.Alberti certainlytook for grantedthe mani- representa particularperson, such as Caesaror Cato. Using festationsof the principle that Cassirerand Panofskycalled these instruments, one can reproduce the totality of the "the homogeneity of space";our question here can only be shape of the object, "the lineaments and the position and whether he assumed, following the Aristotelian tradition, collocation of parts."62The instrumentsAlberti invented to that dimensionsmust belong to physicalbodies or whether perform these jobs are tools to measure and determine the he thought ("with most people," as Aristotle would have shape of the object in order to reproduce it subsequently. said) that places and space can exist independently of the Once the prototype has been properlymeasured and all the physical bodies they contain. datawritten down, it is possible to produceparts of its copy in differentplaces-such as Luni and Paros-and then put Alberti's of them together at a third place.63 Concept Spatium A very specific aspect of these considerations, which A number of contexts and statements in Alberti'swritings certainlyinfluenced Alberti to explore the quantificationof allow us to conclude that he postulated space as an imma- spatialrelationships, has to do with the problem of how to terial entity and that the program of quantification communicateabout architecturalshapes. An extensive dis- describedabove was not related to an Aristotelianposition. cussion of this problem in Alberti has been provided by Joan Gadol remarkedabout Alberti'streatment of perspec- Mario Carpo.64On the one hand, Alberti was extremely tive: "Visualphenomena, be they objects to be painted or concerned with the shapes and formal properties of works the visualrays by which they are seen, are all treatedin Della of architecture and the visual arts ("lineaments").On the pittura as sensible instances of mathematicalideas"66 and other, he knew only too well how unsuitabledrawings could argued that "in Alberti's mind, an abstract, quantitative be for the transmissionof such considerations.Vitruvius's notion of space had supersededthe mythic notion of space own drawingswere lost, scribeswere notoriouslyunreliable as qualitativeaggregate of places to which different emo- when it came to the reproduction of visual material, and, tional values adhere."Gadol related Alberti'sunderstand- until printing presses started reproducing drawings ing of space as homogenous to his ratherbemused account mechanically, the problem was irresolvable: shapes were in De re aedificatoriaof the vulgar opinion that a picture of infinitely easier and better transmittedby drawings,while God or a saint will hear prayersof votaries if placed in one the only reliable method for wide transmissionof written place but not at another.67 When this problem arises in

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This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions church planning, Alberti is ready to yield to the popular An argument can be attempted that Alberti's use of opinion, but he cannot help finding it bizarre: a prayer in his spatium in such cases corresponds to the Aristotelian "place" view, we are left to conclude, is a cause of physical events and not to our modern "unit of space"-that is, that spatium like any other; if the act of prayer is properly performed, it in these cases is a (non-standard) translation of Aristotle's should work equally well anywhere. topos,and that Alberti relied on the Aristotelian concept of A clear indication of Alberti's awareness that his views of place but used the term spatium instead of locus.This would space are different from those of standard Aristotelianism is lead to the argument that Alberti was merely thinking about an implicit reference from De pictura to the fragment from places along the traditional Aristotelian lines. However, Aristotle's Physics described above-where Aristotle argued there are two contexts in De re aedificatoriain which Alberti's that the immaterial diastemabetween the extreme surfaces of use of the term spatium contradicts the standard Aristotelian a place cannot be the place itself. (As we have seen, this was understanding of toposand shows that he was indeed talking Aristotle's third candidate for the concept of place.) In the about parts ofspace in our modern sense and not Aristotelian medieval translation Aquinas worked with, for instance, this places. The first is Alberti's mention of the way the Sun diastema became spatium. One should be careful not to iden- draws up vapors from the earth and gathers them together tify spatium in this case with the concept of space: spatium can into clouds in the spatium orbis, the space of the world.74 In be merely "dimension."68 It is indeed used that way in the this case, spatium could not possibly be understood or trans- next sentence of the same translation, which presents Aris- lated as Aristotelian topos.Aristotle explained in the Physics totle's description of the fourth view that place is the surface, that the world is not a collection of spatial segments and and argues that there can be no immaterial extension between that places are not merely parts of a bigger place that points of the surface that would not belong to the infill of the encompasses them and constitutes the world.75 (This is pre- place.69 Paragraph 30 of the second book of Alberti's De pic- cisely why Aristotle is said not to have had the concept of tura, however, makes a statement that explicitly opposes this homogenous space: places are not merely definable by their Aristotelian account: a painter depicts "the space of the place" position in the wider "place" that is the world.) (huiusloci spatium). The point could not have been made more The second context showing that Alberti's use of the clearly: a place contains space independently from the body word spatium cannot be understood as Aristotelian toposis that fills it. Alberti's choice of terminology here points to the his discussion of the power of the river when it goes from Latin version of the Physics.Elsewhere in De pictura, spatium narrow canals into wider spaces, from "faucibus angus- indeed sometimes means "dimension," but in this context tioribus"into "spatia laxiora."76In this case too, the "spatia" such a translation simply would make no sense.70 into which water flows cannot be understood as Aristotle's It is thus proper to ask how Alberti used the term topoi-according to Aristotle's Physics, a river is a topos in spatium. In spite of Peter Collins's claim that the word did itself; particles that move down a river do not move from not appear in architectural writings before the eighteenth one place to another, but within the same place.77 For this century, it appears ninety-eight times in Alberti's De re aedi- reason, Aristotle argued that in the case of a boat going ficatoria. In twenty-eight cases, the term is merely used to down a river, the place is the whole of the river and also the denote distance between two objects." This corresponds to vessel in which the boat is contained: it is the river, as the the ancient meaning of the word. Aristotle's diastema(but not permanent and stable whole which is the place of the boat. topos!)was traditionally translated this way. There are also When Alberti thus said that water comes into "spatia lax- three contexts in which Alberti used the term to refer to a iora," he could not have possibly understood these "spatia" period of time, again a meaning any standard Latin dictio- to be Aristotle's topoi. nary will mention.72 But in the remaining sixty contexts (in A possible counterargument to the interpretation that some of which the word appears more than once), the word would see in Alberti's statements the assumption of better translates as fully equivalent to our modern "space" or homogenous space derives from a tendency to identify the "part, segment, unit of space." In the latter sense, Alberti's idea of homogenous space with the idea of infinite worlds. usage corresponds to the way we talk, for instance, about One might try to argue that a fifteenth-century author like rooms as "spaces.""73Alberti used the term in this way partic- Alberti could not have subscribed to such a view. When it ularly often-after all, a treatise on architecture needs to dis- comes to perspective, the understanding of the world as cuss the spaces of which buildings are planned and composed. finite or infinite can be seen as closely related to the defin- Spatium consequently appears on those occasions when we ition of vanishing point. It is, however, uncertain that this would say "a space" to refer to a room, landing on a staircase, counterargument can be pushed very far. Alberti was niche in a wall, and so on. remarkably vague when it came to the question of whether

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This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions the vanishingpoint stood for infinityor not. In Depicturahe lacy reduces a historian'sjob to one of the classificationof wrote that the lines going to the central point showed the individualsaccording to collectives ("traditions")to which change in the horizontal lines parallelto the picture plane the historianbelieves they belong. almost to infinity.78 It was in order to avoid this pitfall that in the presen- In fact, the difference between homogenous and het- tation of Alberti'sviews I have intentionally avoided refer- erogeneousspace is not necessarilyrelated to the problemof ences to other historical sources except Alberti's own whetherspace is conceivedas finite or infinite.Homogenous writings. It was necessary to let Alberti speak for himself. spacecan be definedand conceivedof both as finite and infi- While he would have been aware of the Aristotelianposi- nite. With some adjustments,Cassirer's postulate of homo- tion, specificdetails of his argumentsshow that he relied on geneity can be takento imply both finite and infinitespace.79 the concept of the homogeneity of space. His use of termi- It is possible to imagine both infinite and finite space that nology clearly shows that he knew when he needed to dis- would consist of points defined exclusivelyby their geomet- tance himself from Aristotle: he carefully introduced the ricalposition and havingno other qualityon their own. Nor concept of the "space of the place" and he used the term should one easily assumethat the idea of the infinite world spatiumin a way that was equivalentto our modern sense. was inconceivable for ancient, medieval or Renaissance It is important to point out here that the concept of authors.8"Indeed, the idea of afinite world was condemned space as an immaterialthree-dimensional medium in which by the Church as early as 1277: had God created only our objects are located was not something that became conceiv- world, with no positive places beyond, it would follow that able only in the yearspreceding the discoveryof perspective not even He could move the sky in rectilinear motion or in post-Renaissancetimes. Alberti'sviews were neither becauseno positivelycreated places would exist to receive it novel nor original.We have seen that Aristotledescribed the and because the sky would in that case leave a vacuum idea, althoughhe dismissedit. During the Renaissance,the behind.81 This view was condemned because it would conceptwas associatedwith the sixth-centuryChristian Aris- delimit God's power.As for Alberti, he was certainlyaware totelian commentator Philoponus (John Grammarian).83 of the Epicureanidea of the infinite void filled with infinite Philoponus understoodplace as an extension in space;spa- worlds, becausehe mentioned it in the Momus.82Even if we tial extension in his view is a three-dimensional expanse could resolve the question of whether Alberti believed in a which is immobileand containsbodies. 84 Philoponus explic- finite or infiniteworld, this would not tell us anythingabout itly stated that the space (hora)of a container is not to be his views on the homogeneity of space. identified with the matter contained.8"The extension betweenthe boundariesof a container(diastema) cannot exist independentlyof the bodies it contains(a vacuumis not pos- Historical Considerations sible) but it is still an entity in its own right, differentfrom All historical research endeavors to interpret sources in the body contained.86There is no reason to assume that their context. However, it becomes a problem if the histo- Philoponus believed that in the space he described there rian's preconceptions about the context determine the existed dimensions that could not be compared, so his understanding of the content in such a way as to dismiss descriptionis one of homogenous space. Indeed, he further anything that could contradict these preconceptions.This developedthe idea preciselyin that direction.He criticized problemcould be named "the collectivistfallacy," and mod- Aristotle'sview that places can have power on their own- ern scholarshiphas inherited it from Romanticist histori- that placesbelow, for example,attract heavy and places above ography. It reduces individual authors and their views to attractlight things."87Things, saysPhiloponus, simply desire mere manifestations of a predetermined narrative into to be in the stations the Creatorallotted them and there is which they have to fit; how they fit determines their no need to postulate separatepower of places. Philoponus "importance."The result is a methodology that does not also answeredthe potential criticismthat such homogenous allow authors'ideas to be consideredindependently of what space would imply infinity.The argumentof the critics,he we conceive to have been the typical views of their con- said, would be that such an immaterialspace would have no temporaries or the collective under which we subsume boundary,because the conceptof boundarybelongs to mate- them. Psychoanalysis teaches us that the defense mecha- rial bodies. Philoponus's answer is that the space he has nisms of narcissistic patients reduce their capacities for described can subsist only as the place of materialbodies, understanding other human beings as individuals; such and only so much of it as there are materialbodies to fill it. patients alwaysneed to subsume others under a limited set 88 Consequentlyits limits must coincide with those of the of alreadyavailable narratives. Similarly, the collectivistfal- materialworld.

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This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions Philoponus was not alone in his criticism of Aristotle's problem is equivalent to but not identical with the one on views, and in classical antiquity similar views were expressed which Panofsky's understanding of Aristotle stumbled: how by Theophrastus, Strato, and Simplicius.89 Outside the Aris- can things have comparable dimensions without being in a totelian tradition, the view of space as homogenous was space in which these dimensions can be compared? If developed, for instance, by Epicurus, in his Letter to Herodotus, Panofsky's mistake was that he found Aristotle's ontological which had wide circulation in the early Renaissance since it parsimony and the rejection of immaterial things difficult was part of Diogenes Laertius's Lives of Philosophers,trans- to accept, he is, one must note, in the best company one can lated into Latin by Ambrogio Traversari.90Among medieval imagine. Although he was wrong about the issues of the his- authors, Roger Bacon's definition of vacuum as an extension tory of philosophy, his intuition did not fail him in matters that has dimensions but contains no body also belongs to this of art history: as we have seen from Alberti, it was not the kind of position.91 During the Renaissance, Philoponus's view Aristotelian understanding of places and homogeneity with- on space was popularized by Gianfrancesco Pico della Miran- out space but the proper homogeneity of space that stood at dola; the Greek version of Philoponus's commentary on the the beginning of Renaissance perspective. Alberti, we have Physicswas printed in 1535; the first Latin translation came seen, operated with the concept of homogenous space: he out in 1539 and there was one more in 1558.92 The com- postulated an entity he called spatium that can be depicted mentary was printed ten times during the sixteenth century.93 independently of the bodies that fill it. All relationships Because Philoponus's views of space and vacuum as well as between points in spatium are quantifiable and geometri- his theory of impetus as the cause of movement corresponded cally definable; otherwise there would exist shapes that one to those of the newly emerging science, one should not be could not represent in perspective. From every point in surprised that he is among the most cited authors in the spatium, it is possible to draw identical figures-otherwise it works of the young Galileo.94 Philoponus's commentary on would be impossible to reproduce Roman capitals elsewhere the Physicswas also not unknown in the quattrocento: Cardi- or produce parts of sculptures in Paros and Luni and put nal Bessarion owned a copy and another seems to have been them together at a third place. In other words, Alberti's available in Florence as well.95 Philoponus's idea of space spatium conforms to Cassirer's definition of homogenous would also have been known secondhand. Medieval scholas- space. The idea of space as an immaterial, three-dimen- tic philosophers, for instance, had learned about Philoponus's sional, and homogenous medium in which material objects views through Averroes's commentary on the Physics.96 The are placed was widely known in his time, although not in idea was thus certainly not inconceivable. Indeed, from every- line with the standard Aristotelianism. Aristotle's Physicscer- thing we know about Alberti, it is far more inconceivable that tainly exercised a huge impact on Renaissance thinkers, but he did not know about it.97 But even without Philoponus, the the philosopher's teaching about places and space had been idea about space as an immaterial three-dimensional medium controversial since classical antiquity, and most likely is commonsensical.98 Edward Grant has described in his his- Alberti was aware of it. tory of the concept of vacuum that during the Middle Ages, the idea of immaterial three-dimensional space was com- monly referred to as "the vulgar opinion"-for instance, by Notes Pseudo-Siger of Brabant, John Buridan, and Albert of Sax- I should like to expressmy gratitudeto the HarvardUniversity Center for The idea was thus not and of mainstream ony.99 part parcel RenaissanceStudies "I Tatti"; Canadian Centre for Architecture;Humboldt Aristotelianism, but it was certainly known and conceivable. Foundation; Technische UniversitditBerlin; and my home institution, Nevertheless, none of this contextual consideration proves Unitec Instituteof Technology,for the supportthat has enabledme to work The CanadianCentre for Architecturehas also anything about what Alberti thought we should read from on this project. kindlyper- mitted the reproductionof two illustrationsfrom treatisesby Palladio and his own writings. Vignola.My specialthanks to SamuelEdgerton, Christopher Martin, Peter Lautner,Mario Carpo,Stojan Rebid, Steve Wassel,Tony van Raat,Richard and SamirYounds for and advicewith the numerous Conclusion Anderson, help prob- lems I have faced while writing the article, to Karen Wise for help with Etienne Gilson, in his Being and Some Philosophers,imagined written English, and to Peter McPherson for technical help in the prepa- ration of the I an abbreviatedversion of the article that Plato lived long enough to read the first book of the manuscript. presented at the conferenceof the Society of ArchitecturalHistorians of Australiaand Metaphysicsand then composed a dialogue titled Aristotle.100 New Zealandin Melbourne in September2004. In the dialogue, Socrates asks the young Aristotle: "Then, my lad, I wish you could tell me how it may be that beings 1. Aristotle, Physics,208b31-209al, in Jonathan Barnes,ed., The Complete are, through sharing an essence, which itself is not." The WorksofAristotle: The Revised Oxford Translation (Princeton, 1984). All cita-

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This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions tions of Aristotle in English translationare from this publication.For the "AestheticFormalism in RenaissanceArchitectural Theory," Zeitschriftfiir originalGreek version of this passage,see n. 35. Where it is necessaryto cite Kunstgeschichte66, no. 3 (2003), 321-40. the Greek original,this is done accordingto the Loeb edition. 9. Alberti's De pictura, De statua, and Elementa picturae are cited according 2. ErwinPanofsky, "Die Perspektiveals 'symbolischeForm,"' in Fritz Saxl, to the parallelGerman-Latin edition of Leon BattistaAlberti, Das Stand- ed., Vortraigeder Bibliothek Warburg 1924-1925 (Leipzig and Berlin, 1927). bild, Die Malerei, Grundlagen der Malerei, ed. Oskar Bditschmann (Darm- All quotationsare from a recent reprintin ErwinPanofsky, Deutschsprachige stadt, 2000). For Italian versions of the treatises on painting and the Aufsiitze,ed. Karen Michels and Martin Warnke (Berlin, 1998), vol. 2, elementsof painting(with parallel Latin version), see Leon BattistaAlberti, 664-757. See also Erwin Panofsky, Perspective as Symbolic Form, trans. Operevolgari, ed. Cecil Grayson(Bari, 1973), vol. 3, 6-129. For De re aedi- ChristopherS. Wood (New York, 1991). ficatoria,I haveused Leon BattistaAlberti, L'architettura, parallel Latin orig- 3. The section Panofskycited from Cassireris worth repeatinghere: inal and Italiantranslation, ed. and trans. Giovani Orlandi(Milan, 1966). See also the English translation:Leon BattistaAlberti, On theArt ofBuild- Die Homogenitatdes geometrischen Raumes beruhtletzten Endes darauf, ing in TenBooks, trans. Joseph Rykwert,Neil Leach, and Robert Tavernor dali alle seine Elemente, dal die ,Punkte', die sich in ihm zusammen- (Cambridge,Mass., 1988). I have also used Javier Fresnillo Nsifiez'sLeon schlielen, nichts als einfache Lagebestimmungensind, die aber auserhalb Battista Alberti: De re aedificatoria A Lemmatized Concordance(Hildesheim, dieser Relation,dieser ,Lage',in welcher sie sich zueinanderbefinden, nicht 1996), which gives pagination accordingto Orlandi'sedition. In order to noch einen eigenen selbstandigen Inhaltbesitzen IhrSein geht in Ihrem facilitatethe use of my citations, I have added, in brackets,page and line wechselseitigenVerhaltnis auf es Istein reinfunktionales, kein substantiales numbersaccording to the 1486 edition, which is more usuallycited. Sein. Well diese Punkteim GrundeOberhaupt von allem Inhaltleer, well sie 10. Elkins,Poetics, 23. zu bloSenausdrOcken ideeller Beziehungen geworden sind, darum kommt es 11. Ibid., 24; Peter Collins, Changing Ideals in Modern Architecture fOrsie auch Gleichartigkeitihrer Struktur, die in der Gemeinsamkeltihrer jene 1750-1950 (London, 1965), 285. logischen Aufgabe, Ihrerideellen Bestimmung und Bedeutung gegrindet 12. Elkins,Poetics, 23. ist . der geometrische Begriffder Homogenitatgeradezu durch das Pos- 13. "Homology"here means that the shape and size of individualelements tulatausgedrickt werden kann,daS von jedem Raumpunkteaus nach allen can be consistentlyrestituted (read) from individualprojections (such as plan, Ortenund nach allen Richtungengleiche Konstruktionenvollzogen werden section, and elevation)according to the geometricalrules for the given pro- k6nnen jection (rules for orthogonal projectionsin the case of plan, section, and Panofsky, "Die Perspektive," 667-68; Ernst Cassirer, Philosophieder symbol- elevation).It is thus not possiblethat one and the same fagadehas two doors ischen Formen, vol. 2, Das mythische Denken (Berlin, 1925), 107. See also in elevation but one in plan, or that the elevation indicates 25 m as the Ernst Cassirer, Individuum und Kosmos in der Philosophie der Renaissance length of the faqade,whereas the plan says that it is 23 m. (Darmstadt,1977), 11, 26, 192-93. For an analysisof the structureof Panof- 14. AlbertoP6rez-G6mez and Louise Pelletier,Architectural Representation sky'sreformulation of Cassirer'sdescription of differenttypes of space, see and thePerspective Hinge (Cambridge, Mass., 1997), 26. The authorsdo not JamesElkins, ThePoetics ofPerspective (Ithaca and London, 1994), 190-201. specify the natureof their insight into "Brunelleschi'sexperience." For Cassirer'sinfluence on Panofsky,see Michael Ann Holly, Panofskyand 15. Ibid., 98. the Foundations ofArt History (Ithaca, 1984), 114-57. 16. Ibid., 21. 4. This was also Cassirer'sview; see Individuum,192. Cassirerascribed the 17. As Martin Kemp remarkeddiscussing the argumentthat Brunelleschi firstphilosophical formulation of the concept to Cusanus(ibid., 11, 27) and did not constructthe Baptisterydrawing geometrically but producedit by seems to have been genuinely unawareof Philoponus'sdiscussion of the copying a mirrorimage: concept in the commentaryon the Physics. The most serious objectionto the idea of Brunelleschipainting on a mirroror 5. Samuel Edgerton, The Renaissance Rediscoveryof Linear Perspective (New "copying"a mirrorimage is that such a proceduredoes not explicitlyembody York,1975), 161. perspectiveas a conscious process of geometricalconstruction. A painting 6. "Heterogeneous"is used here to denote a space that is not homogenous made by these means mightachieve a highdegree of naturalism,if the tech- accordingto Cassirer'sdefinition. nicalproblems could be overcome, but it would not be "partof that science" 7. The relationshipbetween the Renaissancedevelopment of perspective calledperspective which sets thingsdown "Inthat misurawhich corresponds as a method of geometrizationand quantificationof visual perception and to the distance from which they are shown" (Manetti) Judgment on this the developmentof the theory of the classicalorders as a method of quan- question is dependent not upon the detailed Interpretationof Manetti's tificationof architecturalelements has been little exploredso far.Like per- account, but upon acceptance or rejectionof its whole basis If we accept spective,the theory of ordersnecessarily implies the homogeneity of space: that Brunelleschi'smethod was perspectival,we must believethat it involved if it were not possible to draw the same figure from every point in space, a process of greaterrigour and more explicitgeometry than the copyingof a then it would not be to carve the same in certainly possible capital every mirrorreflection. point of space. David Summers has insightfully drawn the parallel between Brunelleschi's experiments on perspective and his restoration of the classi- Martin Kemp, "Science, Non-Science and Nonsense: The Interpretation of cal orders. David Summers, Real Spaces: WorldArt History and the Rise of Brunelleschi's Perspective," Art History 1 (June 1978), 134-61, 148-49. WesternModernism (New York, 2003), 513. Starting from the assumption of 18. P~rez-G6mez and Pelletier, Architectural Representation,27. heterogeneous space, the sixteenth-century development of the complex 19. See n. 59. systems of quantification of architectural shapes in order to replicate them 20. Because the authors have used here the word "homology" without defin- (as described recently by Mario Carpo, "Drawing with Numbers: Geome- ing it previously, one may be tempted to wonder whether their statement try and Numeracy in Early Modern Architectural Design," JSAH 62 [Dec. could mean something else. Shorter Oxford English Dictionary, vol. 1,1261, 2003], 448-69) would not have been possible. states that "homologous" is "used of elements (lines, points, terms, and so 8. On the question of whether Renaissance architectural theorists could forth) having similar or analogous positions or role in distinct figures or have conceived of purely formal aesthetic judgments, see Branko Mitrovid, functions." Insofar as a plan and elevation of a building are "distinct fig-

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This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions ures,"Perez-G6mez and Pelletier'sstatement must mean that Brunelleschi 33. Aristotle,Physics, 213a12-218a20. was not able to understandthat the position and length of a line in eleva- 34. Ibid., 215a24-215b22. tion must be analogousto the length and position in the plan. 35. Ibid., 208b31-209a3. Aristotlesays that Hesiod'sview was derivedfrom To 21. Ibid., 5. the belief "3s 8iov Trp(TovvU1TdpatL XCXpayv To3 ot010, 8tS6 VOL~tLELV, OL continued: 8' 22. In "Perspektive,"Panofsky assumed from the beginning that perspec- (aOTrEp TrokhoL,TrdvTCa ELvaO TroU KOLiV TO6Tr."Aristotle "EL tive necessarilyimplies a certainRaumanschauung (664), that a perspectival EiTL TOLODTO, 00UvCLTflTi L1I EI flTOTOO T6ovo 8VVtCtI KL NTTPOTEpCaTrdv- TL" TWV picture is a window through which we believe we look into a Raum(664), Ol yp iVEVUTWV XXi oOVEV EOTLv ta , KEiLVO8('vEU dLVyKT) TO*)0, ,CXX0V, and that Gesamtraumis projectedon the picture plane (665) (see n. 2). The TrpWTov Oi yap T6a'6 TOTO!V vaVCOTId 1OEtpoLivbov." The LVtL" 5Trr6XXUatL definitionof perspectivehe suppliedstates that it representsobjects with a implicationof the first sentence is that T6TrOSis conceived as part of Xw'pa. part of space (665 n. 5). One should note that the introductionof the con- See Keimpe Algra, ConceptsofSpace in Greek Thought (Leiden, 1995), 124-25, cept of space is in fact gratuitous.In all these contexts, Panofsky'sstate- for the relationshipbetween these two termsin Aristotle. ments could be reformulatedin a way that would not postulatespace. Raum 36. Ibid., 209a5-209a30. could be systematicallyreplaced with the "relationshipsbetween shapesand 37. Aristotle, Categories,2a35-2b6. dimensions of bodies." In such a nominalist reformulation, dimensions 38. For Philoponus's views on space, see loannis Philoponi in Aristotelis Physi- would alwaysbe propertiesof materialobjects (includingair, in the case of corum libros quinque posteriores commentaria (Berlin, 1888), esp. "Corollar- distancesbetween bodies) and there would be no need for an immaterial ium de loco," 556.25-585.4. A recent English translationof this important entitysuch as space.This is important,because when Panofskylater pointed section is in the samevolume with Simplicius,Against Philoponus on theEter- out that the ancient philosophersand artistsdid not have a concept of sys- nity of the World, in Philoponus, Corollaries on Place and Void, trans. David tematicspace similarto his and Cassirer's(699), this still did not constitute Furley and ChristianWildberg (London, 1991). A useful introductionin a validreason to claim that they thereforecould not have developedthe geo- Philoponus'sphilosophy is JohannesPhiloponos, Grammatikosvon Alexan- metricalconstruction of perspective.His accountin this section (699-70) is dria. Ausgewdhlte Schriften, introduction and commentary by Walter Bohm delimitedto very brief mention of the views of Democritus,Plato, andAris- (Munich, 1967). For the theory of space, see esp. 79-96. totle. In the case of Aristotle, he presented accuratelyAristotle's theory of 39. Philoponus, "Corollarium,"578.5-25. place as the containerof the body and arguedthat Aristotle'sview on infin- 40. It is interestingto thinkthat the introductionof the idea of homogenous ity precludedascribing to Aristotlethe understandingof space as homoge- space in the Aristoteliansystem had to have Platonizing implications.In a nous (670). However, for his argumentto be valid, it is reallyhomogeneity note in his English translationof this section in Philoponus, David Furley and not the understanding of space that matters. Showing that ancient noted that the term "selfhypostasized" or "selfconstituted, "a-oOBv60VrTaTO" philosophersoperated without the concept of spacedoes not mean that they came directly from Neo-Platonist sources, such as Proclus. Philoponus, could not have had understoodthe relationshipsbetween the dimensionsof Corollaries,trans. Furley and Wildberg, 40. (Cf. Proclus, The Elementsof bodies (includingall manifestationsof the homogeneity of space) in a way Theology,Greek originalwith English translationby E. R. Dodds [Oxford, that could have enabledthe constructionof perspective. 1999], 43-51.) 23. For a presentationof Aristotletheory of space, see Helen S. Lang, The 41. Aristotle, Categories,6a26-28. TE Order of Nature in Aristotle's Physics (Cambridge, England, 1998). 42. He is very explicit that "EKaorTOvyTp TOV ELtpfLivUvIroa1yv 0aov KL01 24. Aristotle,Physics, 208b8 (see n. 1). dtvoov Aristotle, Categories,6a27. This would not be possible if Xy•Tyat." " ( side from which 25. Aristotle wrote: 'akX oljK iV YLVOVT1L iS3 i•pOS iE•t TO0 the lengths of lines depended on their locations or on the T6T,0P, The idea seems measured. T6rrov60EiTt T6L OST XoOU TOfOopavoO" (ibid., 211b29). they are to be that it simply makesno sense to try to determinethe spatialposition 43. These figures need not be delimited by the size of an individualplace of a place in relation to other places in which it is contained,including the and may extend over any number of places. One and the same line can be world itself. drawn across a wall and a painting on the wall, thus extending over places 26. Ibid., 209b6. (that is, accordingto Aristotle'sdefinition, external surfaces) on the wall and 27. This is Aristotle's account of Plato's position. See Aristotle, Physics, the painting.This line can be equal in length to a line drawnon the floor. 209b13. Aristotle'simportant point is that one can account for this without needing 28. Ibid., 210a5. the concept of space. Panofskysays that for the ancients, the world always 29. Ibid., 211b13. remaineddiscontinuous ("stets bleibt das Ganze der Welt etwasvon Grund 30. Lang, The Order, 87-88. aus Diskontinuierliches"[Panofsky, "Perspektive," 699] [see n. 2]), but in 31. Aristotle,Physics, 212a5. See also Lang, The Order,83-121. fact, Aristotle'stheory of places was developed precisely in order to avoid 32. SeeJonathan Barnes, The Presocratic Philosophers (London, 1982) 402-5, discontinuity,empty space, or vacuum. for a discussionof the problemin the worksof atomists.As an inspiredarti- 44. Aristotle, De anima,419a16-22 (see n. 1). cle in the Encyclopediaof Philosophywarns us, empty space is nothing, and 45. On St. Antoninus, see Samuel Edgerton, The Heritage of Giotto's Geom- saying that it exists equals saying that a non-existing thing exists, which etry:Art and Scienceon the Eve of the ScientificRevolution (Ithaca, 1991), 103-4; eventually threatens to burden ontology with "centaurs and unicorns, car- and Leo Steinberg and Samuel Y. Edgerton, "How Shall This Be?," Artibus nivorous cows, republican monarchs and wife-burdened bachelors"; "ever et Historiae 7 (1987), pt. 2, 46. I am indebted to Samuel Edgerton for draw- since Parmenides laid it down that it is impossible to speak of what is not, ing my attention to St. Antoninus's writings and in particular to the section broke his own rule in the act of stating it, and deduced himself into a world discussed here. where all that ever happened was nothing, the impression has persisted that 46. See Sancti Antonini Summa Theologica (Verona, 1740; facs. ed. Graz, the narrow path between sense and nonsense on this subject is a difficult one 1959), vol. 1, 122. to tread and that the less said of it the better." Peter L. Heath, "Nothing," 47. Aristotle, De anima, 418b10, De sensu, 446b27 (see n. 1). in Paul Edwards, ed., The Encyclopediaof Philosophy(New York, 1967), vol. 48. David Lindberg, Theories of Vision from Al-Kindi to Kepler (Chicago, 5,524. 1976).

LEON BATTISTA ALBERTI AND THE HOMOGENEITY OF SPACE 437

This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions 49. Ibid., 152. (182v.16), 921 (186.33), 971 (197.10), 979 (198v.24). 50. Samuel Edgerton, "Alberti's Perspective: A New Discovery and a New 72. Ibid., 143 (30v.1), 937 (190.4), 963 (195.29). Evaluation," Art Bulletin 48 (Sept.-Dec. 1966), 367-78, 373. 73. Ibid., 21 (4v.20), 23 (5.13), 27 (6.2), 53 (11.26), 53 (llv.2), 53 (11v.9-12), 51. For different approaches to the functioning of sight and especially debates 85 (18v.11),87 (18v.31),89 (19.27), 91 (19v.16),99 (21.26), 167 (35.21), 289 between intromission and extramission theories, see Lindberg, Theories. (59v.29),291 (60.9), 327 (68.19), 343 (71v.10),357 (74v.14),365 (76.2), 367 52. Ibid., 149. (76.32), 373 (78v.26), 379 (79.31), 381 (79v.4), 399 (83v.17), 407 (85.22), 53. "Visum per triangula fieri cuius basis visa quantitas cuiusve latera sunt 415 (87.15),417 (87.21),417 (87v.8),433 (90v.21),433 (90v.30),461 (96.25), iidem ipsi radii qui a punctis quantitatis ad oculum protenduntur." Alberti, 497 (103.32), 509 (106.19), 533 (111.4), 533 (111.19), 553 (115.31), 591 De pictura, 1.6 (see n. 9). (122.17), 593 (122v.10), 599 (123v.1), 605 (124v.6), 605 (124v.22), 607 54. Ibid., 1.2. (125.5), 627 (129.10), 639 (131.24), 645 (132.21), 647 (132v.21), 651 55. Ibid., 1.7. (133.33), 687 (141v.16), 707 (144v.24), 707 (145.2), 713 (146.23), 715 56. Ibid., 1.12. (146v.4),719 (147.32), 723 (148.3), 737 (150.30), 753 (153v.2),753 (153v.8), 57. Ibid., 1.12, 1.23. 775 (157.14), 783 (158v.29), 791 (160.6), 793 (160v.9),949 (192v.11),977 58. Ibid., 1.19. (198v.4). 59. "Denique meministi quae de pavimenti parallelis et centrico puncto 74. Ibid., 27 (6.2). atque linea diserui. In pavimento ergo parallelis inscripto alae murorum et 75. Aristotle,Physics, 211a27-29. quaevis huiusmodi, quas incumbentis nuncupavimus superficies, coaedifi- 76. Alberti, De re aedificatoria, 949 (192v.11). candae sunt. . ... Principio ab ipsis fundamentis exordium capio. Lati- 77. Aristotle,Physics, 212a 17-22. tudinem enim et longitudinem murorum in pavimento describo, in qua 78. "Demonstrant quemadmodum paene usque ad infinitam distantiam quidem descriptione illud a natura animadverti nullius quadrati corporis quantitates transversae successivae sub aspectu alternantur." Alberti, De pic- rectorum angulorum plus quam duas solo incumbentes iunctas superficies tura, 1.19. It has been denied by some scholars that Alberti had the con- uno aspectu posse videri." Ibid., 1.33. cept of a vanishing point, since he used the term "central point." Elkins 60. "Ex his quae sequentur, omnis ratio et via perscribendi componendique thus suggested that "the concept of a vanishing point was exposed only in lineas et angulos et superficies explicabitur notaque reddetur adeo ut nihil 1600" (Poetics, 8), but later in the book he referred to "Alberti's vanishing in rerum natura sit, quod ipsum oculis possit perspici, quin id hinc instruc- point" (145) (see n. 3). tus perfacile possit lineis perfinire atque exprimere." Alberti, Elementa pic- 79. The formulation that in homogenous space it should be possible to draw turae, E (see n. 9). What follows ("quae sequentur") is a list of twenty-five the same figures from every point suggests infinite space: if we draw a fig- various simple geometrical constructions. ure from point A so that Al is the point on the figure which is the most 61. Alberti, De statua, 7, 11 (see n. 9). remote from A, then draw the same figure from Al so that A2 on that fig- 62. "Lineamenta et partium situs et collocations." Ibid., 6. ure is the point which is the most distant from Al, we can extend this 63. Ibid., 6, 16. See also Mario Carpo, "Ecphrasis geographique et culture process ad infinitum and the result will be an infinite homogenous space. ' visuelle I'aube de la revolution typographique," in Leon Battista Alberti, But one could formulate an equivalent homogeneity postulate by saying Descriptio urbis romae, ed. Martine Furno and Mario Carpo (Geneva, 2000), that in a homogenous space, it is possible to draw the same figures from 65-96, esp. 91. every point except in the case that a line belonging to the figure exceeds 64. Mario Carpo: "How Do You Imitate a Building that You Have Not the distance between the point and the end of space in the direction of that Seen? Printed Images, Ancient Models and Handmade Drawings in Renais- line-and the result will be a finite homogenous space. sance Architectural Theory," Zeitschriftfiir Kunstgeschichte64, no. 2 (2001), 80. A particularly useful presentation of arguments used through history is 223-33, esp. 225; see also Carpo, "Ecphrasis." in Richard Sorabji, Matter, Space and Motion: Theoriesin Antiquity and Their 65. Alberti, De re aedificatoria, 575 (119v). Sequel (London, 1988), 125-202. 66. Joan Gadol, Leon Battista Alberti: Universal Man of the Early Renaissance 81. See Heinrich Denifle and Emil Chatelain, eds., Chartularium universi- (Chicago and London, 1969), 28. tatis Parisiensis (Paris, 1889; repr. Brussels, 1967), vol. 1, 546. For a discus- 67. Alberti, De re aedificatoria, 661-63 (135v.30-34). sion of the condemnation, see Edward Grant, Much Ado about Nothing: 68. Aristotle wrote: TIT To' aT~iV TJOV (Physics,211 b8 Theories of Space and Vacuumfrom the Middle Ages to the Scientific Revolution "tdorT•ld Ei(XadTv" [see n. 1]), which in Aquinas became "aliquod spatium inter extrema conti- (London, 1981), 324 n. 29; and Jan A. Aertsen, "Zur Einleitung," in Jan A. nentis." Thomas Aquinas, "In libros physicorum," in Robert Busa, ed., S. Aertsen and Andreas Speer, eds., Raum und Raumvorstellungen im Mittelal- Thomae Aquinatis Opera Omnia (Stuttgart, 1980), vol. 4, 052 CPY lb4, lc6, ter (Berlin, 1998), xiii. The latter discussion also includes the relevant bib- n. 2, 87. liographical references. 69. Aristotle said that the fourth option is that place is "Ta JXaTt a EL'[l 82. Leon Battista Alberti, Momus (Cambridge, Mass., 2003), 3.16, 4.13. Epi- EaTt 8ltdcCTrLanrrapc TO TOO ot no~S Clye•0oS." Aris- curus's views on infinity are revealed in his Letter to Herodotus. See Hermann s•L8nV TyyLvovivov totle, Physics, 211b9. In Aquinas, this became "si nullum spatium est inter Usener, Epicurea, cited according to the parallel Greek-Italian edition, Ilaria extrema continentis, quod habeat aliquas dimensiones, praeter magni- Ramelli, ed. and trans. (Milan, 2002), 41-42. tudinem corporis quod ponitur infra corpus continens." Busa, S. Thomae, 83. On Philoponus, see Richard Sorabji, ed., Philoponus and the Rejectionof vol. 4, 052 CPYlb4, lc6, n. 2, 87. Aristotelian Science (London, 1987). I am indebted to Peter Lautner for 70. See Alberti, Depictura, 1.6 (see n. 9). drawing my attention to the importance of the Renaissance reception of 71. Alberti, De re aedificatoria, 59 (12v.29), 79 (17.29), 87 (18v.22), 297 Philoponus in this context. (61v.9-10), 299 (61v.16), 311 (64v.6), 347 (72.26), 397 (83.4), 421 (88.18), 84. On Philoponus's views on space, see David Furley, "Summary of Philo- 477 (100.18), 483 (100.23), 491 (102v.21), 557 (116.4), 571 (119v.33), 573 ponus' Corollaries on Place and Void," and David Sedley, "Philoponus' Con- (119.21), 597 (123.17), 623 (128.30), 709 (145v.18), 713 (146.13), 735 ception of Space," in Sorabji, Philoponus,130-39 and 140-53, respectively. (149v.16), 751 (153.13), 753 (153v.5), 763 (155.25), 837 (169v.17), 903 85. Philoponus, "Corrolarium," 568.25-569.7 (see n. 38).

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This content downloaded from 64.9.76.165 on Mon, 10 Feb 2014 11:35:15 AM All use subject to JSTOR Terms and Conditions 86. The section clearly identifies XW'pawith 8dOTTLGa:"XEL'TTaL Spa TT'V secundum eos." Roger Bacon, Opus majus (Oxford, 1897), vol. 2, 525. ( 92. On the Renaissance of see Charles XC)pavTflV pTa~ri TaOV TEpidTTvTC)V i'VTS TOa1topi•E0 CI ETP?Eptv"EOTL reception Philoponus, Schmitt, in the Sixteenth tpa TL To CIETag?8cioTdrta irap& TdACLTLiTTTOVTa oa(CaTa." Ibid., 569.5-7. "Philoponus' Commentary on Aristotle's Physics Century," 87. Strictly speaking, this view of Aristotle's does not contradict the homo- in Sorabji, Philoponus, 210-30. geneity postulate as defined by Cassirer. It is possible to imagine a homoge- 93. Schmitt, "Philoponus' Commentary," 229. nous space in which one could draw identical figures from every point and 94. Willam A.Wallace, Prelude to Galileo: Essays on Medieval and Sixteenth- in which, nevertheless, points in certain regions would have the power to Century Sourcesof Galileo's Thought (Dordrecht, 1981), 136, 196-97. See also attract a certain kind of matter. But it does contradict the first part of Cas- Schmitt, "Philoponus' Commentary," 225. sirer's definition, which states that points should have no other property 95. Carlota Labowski, Bessarion'sLibrary and the BibliotecaMarciana, Rome (Inhalt, or "content") except their relative position to each other. (Rome, 1978), 191-243; and Berthold L. Ullman and Philip A. Stadter, The 88. Ibid., 582.32-34. Philoponus says: "T61T~oTOCV cWptCtTCOV Public Library ofRenaissanceFlorence: Niccolb Niccoli, Cosimode'Medici and the -r'TETrli," which is a way to avoid saying that nothing exists. The verb is nor- Library of San Marco (Padua, 1972), 257. See also Schmitt, "Philoponus' iVTir•Lt mally translated into English as "to lie underneath," "to be granted, Commentary," 215. assumed," or "to be left remaing." (See n. 32 above.) "Subsist," used by Fur- 96. Averroes's account is brief, barely a couple of sentences, but it does con- ley in the English translation, is equivalent to "subsistere"in Dorotheus's vey the main aspects of Philoponus's position, including his stance on the Latin version: "Cum locus corporum subsitat, tantum equidem subsiteret, theoretical but not real possibility of vacuum. See Aristotelis opera cum Aver- quantum pro recipiendis mundi corporibus opportunum esset." Johannis rois commentariis (Venice, 1562-74), vol. 4, 141. For the reception of Philo- Philoponi commentaria in libros Physicorum interprete Guillelmo Dorotheo ponus's views through Averroes, see Grant, Much Ado about Nothing, 19 and (Venice, 1554; repr. Frankfurt am Main, 1984), 87. esp. n. 61 (see n. 81). 89. See Richard Sorabji, "John Philoponus," in Sorabji, Philoponus, 1-40, 97. Because of the similarity between names, one is tempted to relate the esp. 15 and n. 90 for an extensive list of ancient authors who had similar Aristotelian commentator Philoponus to Philoponius, a personality who views. See also MaxJammer, ConceptsofSpace: The History of TheoriesofSpace appears in Alberti's Intercoenales. See Leon Battista Alberti, Dinner Pieces, in Physics(Cambridge, Mass., 1954), 21. On Theophrastus's critique of Aris- trans. David Marsh (Binghamton, 1987). But every similarity seems to per- totle's position, see Keimpe Algra, Conceptsof Space in Greek Thought (Lei- tain only to the name. On Philoponius, see Mark Jarzombek, On Leon Bat- den, 1995), 231-48. The latter discussion also includes relevant tista Alberti: His Literary andAesthetic Theories (Cambridge, Mass., 1989). bibliographical references. 98. On the presence of the idea in classical antiquity, see Jammer, Concepts, 90. Epicurus, Letter to Herodotus, in Usener, Epicurea, 39-41. For the avail- 8-11. ability of Epicurus in the Renaissance, see Charles Schmitt, ed., The Cam- 99. Grant, Much Ado about Nothing, 9 and n. 1. See also Grant, "Place and bridge History of RenaissancePhilosophy (Cambridge, England, 1992), 781. Space in Medieval Physical Thought," in Peter K. Machamer and Robert 91. "Et voco vacuum quod philosophi posuerunt, scilicet spatium dimension- G. Turnbull, eds., Motion and Time, Space and Matter (Columbus, 1976). atum, non habens corpus locatum, possibile tamen recipere corpora locanda, 100. Etienne Gilson, Being and Some Philosophers(Toronto, 1952), 49-50.

LEON BATTISTA ALBERTI AND THE HOMOGENEITY OF SPACE 439

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