The argument of this essay is a philosophical one, but it has an historical dimension. It starts from a building not usually associated with proportional systems, the Ricetto or anteroom of the Biblioteca Laurenziana in Florence (1525–1534), and a few drawings Michelangelo made in connection with his designs for the San Lorenzo complex (Figs. 1, 2).

What do these images, which combine an architectural profile with a human one, tell us? Do they suggest that the human profile shares some proportional relation with the architectural outline, in the sense of a common pattern of mathematical relationships, for instance that the face can be divided into four equal parts? That the human profile is the visual manifestation of some geometrical substructure? Or conversely, that the geometrical structure of an architectural profile can best be understood in anthropomorphic terms?

Starting from reactions to the Ricetto of the Biblioteca Laurenziana by contemporaries, who tried to make sense of its strange and unprecedented forms either by using the rhetorical concept of compositio or by assuming a proportional system in the vestibule, I will show how in the latter part of the eighteenth century proportion lost its role as the objective foundation attributed to architectural beauty. Instead, beauty became redefined as an experience of the human mind, arising from the accordance between the properties of an object, its sensuous experience and the perceptive apparatus of the human mind. But this redefinition does not mean that proportion, or to be more precise, the assumption of a proportional system, became irrelevant. In the final part of this paper I will argue that in Kant’s aesthetics, proportion, in the sense of a visible set of relations between the dimensions of the parts of a building that can be expressed in mathematical terms, became one of the key features of a building, or indeed any object, that enables the human mind to make sense of, and judge, the objects of sense perception. Continuing Kant’s line of thought I will argue that the assumption of a proportional system, together with the projection of anthropomorphy onto architecture, are the two major hermeneutic strategies by which human beings try to understand buildings. In the course of Western architectural theory there have been other approaches to beauty — Alberti’s definition in terms of splendour and magnificence, for instance — but they lend themselves far less well to conceptualization or theorization than these two, perhaps because of their cultural contingencies.

As is well known, viewers have long found the Biblioteca Laurenziana, and its vestibule or Ricetto in particular, very puzzling.1 In the 20th century Nicolaus Pevsner wrote of the feeling of oppression the room created; James Ackerman of the disobedience to the Vitruvian rules in the orders and the subversion of architectural decorum (Pevsner 1983: 222; Ackerman 1961: 42). More recently Cammy Brothers has argued that the discomfort, disorientation or disquiet experienced by visitors may be accounted for by Michelangelo’s introduction of elements usually reserved for façades into an interior (Brothers 2008: 161–181). Furthermore, it is not a building usually studied for its proportional system.

Michelangelo’s contemporaries, even though they do not report the malaise felt by the historians just mentioned, also found the Ricetto singular and difficult to interpret. Vasari’s comments about Michelangelo’s work in the Sacrestia Nuova, that there he liberated architecture from the chains and bonds by which architects before him had followed a beaten path in the execution of their work, are relevant to this issue, because they give some sense of the metaphors a contemporary drew on to speak about Michelangelo’s work (Vasari 1927, 7: 193). The reactions by Francesco Bocchi, Cosimo Bartoli and Carlo Lenzoni, who drew on analogies between Michelangelo’s freedom in handling the orders and contemporary literary inventiveness, are less well known. Even less studied is the attribution by the 18th-century historian Giuseppe Ignazio Rossi of a proportional system to the Ricetto to make sense of it.

In the Bellezze della città di Firenze, first published in 1591, Francesco Bocchi clearly struggled to make sense of the Ricetto, and is not always successful. After mentioning the very simple measurements of the room (twenty by twenty braccia), he notes that the measurements of the columns (‘le misure’) are different from elsewhere, but their handling here is nonetheless evidence of Michelangelo’s extreme talent. He is also very astute in noting that the brackets, which do not support anything, are handled correctly, because despite appearances, the supporting work is done by the columns.2

Other contemporaries noted the combination of Corinthian proportions with a Doric appearance in the Ricetto’s order: whereas the columns have a Doric capital and base, and no fluting, their ratio of height to diameter is 1 to 9 instead of 1 to 6. Caroline Elam has shown, in her 2005 article on the Biblioteca, that the Florentines Cosimo Bartoli (1503–1572) and Carlo Lenzoni (d. 1551), who, following Vasari, analysed the design in terms of a mixture of Doric ornament and Corinthian proportions, did not make sense of this mixture in mathematical terms (Elam 2005: 61–66). Instead they suggest a parallel with language, and in particular with the question of the composto; i.e., whether it is allowed to make new combinations that did not exist in antiquity, as does the Tuscan dialect, which was seen as a composto of Latin, Etruscan and Tuscan elements. In Carlo Lenzoni’s Difesa della lingua fiorentina, published posthumously in 1556, set in San Lorenzo, and dedicated by its editor Pierfrancesco Giambullari to Michelangelo, the architect is held up as an example and defence of such new developments, in which imitation is replaced by new composti. If critics were to forbid poets to move beyond the Greek and Latin classics and invent something new, architects might just as well limit themselves to the Doric, Tuscan, Ionic, and Corinthian orders, and resign themselves to avoiding the possibilities for new invention afforded by the Composite order (Elam 2005: 64, where this passage is cited in full).

Thus, the novelty of the Ricetto, its strange forms and composto of Doric forms and Corinthian ratios between the diameter and the length of the column, and its novel handling of ornaments such as the brackets, is interpreted in terms derived from the rhetorical concept of compositio. Originally, in Cicero and Quintilian’s treatises on rhetoric, this word was used for composite sentences with subordinate clauses. Subsequently it was introduced into artistic theory by Leon Battista Alberti, to designate the painterly process of putting together the figures of an image into a coherent and persuasive representation, or, the creation of a persuasive unity out of conflicting elements (Baxandall 1971: 129–139; Puttfarken 2000; Van Eck 2007: 66–73, 127–134). Cosimo Bartoli, in his Raggionamenti academici of 1567, draws these strands together: ‘If Michelangelo differed from the Ancients, he has kept a proportion in his works which is very agreeable to the spectator, and gives much pleasure to those who consider it carefully’.3

Another strategy was pursued by Vincenzo Danti, who also dedicated his Trattato delle perfetti proporzioni of 1567 to the artist. Although originally planned to have nine books, only one was finished. For Danti, proportion — that is, proportional beauty as defined throughout this volume — is the visible manifestation of ordine, the commensuration or commisurazione of the parts and the whole, both in the cosmos (i.e., in nature) and the works of man. Such proportion is not limited to quantitative relations. Instead, Danti defines it, in a reprisal of Aristotelian metaphysics, as the beauty resulting from the perfect attezza, aptitude or fitness of the parts forming together a composto ordinato. In other words, proportion here is the expression or manifestation of fitness for the purpose of the whole of the parts of a composto. Proportion redefined as fitness for purpose thereby gives outward form, often based on organic forms, particularly the human body, number, and measure to the parts of a natural organism or human artefact. It also ensures commensurate quantities. In the case of the human body, for instance, the fitness of the members for their functions generates beautiful proportions, and that is what viewers, according to Danti, call beauty. These viewers prefer the composto ordinato, resulting in beautiful proportions of different parts because these offer in their differentiation more variety or varietà. In the human body, the commensurate measurements of the members therefore reflect the underlying fitness of those members. Artists can imitate the composti to be found in nature, in plants and animals, or create their own works and new composti, including the ornaments designed by architects. They can produce new forms which seem to possess much more artifice and perfection than natural organisms; these forms follow architecture’s own laws, but are still governed by the authority of nature (Hemsoll 2003: 61).

In the case of Michelangelo’s Ricetto in San Lorenzo, the basis of his new, composite repertoire of forms is provided by the forms of nature according to Danti. The composti of the architect are combinations of natural forms, mainly derived from the human body, but with an underlying system of commensurate measurements. Such proportional systems therefore, according to Danti, are not just mathematical relations; they are primarily expressions of the fitness for purpose, the teleological unity of the elements of an organism or artefact. In uniting unequal elements, proportional systems create varietà, a major cause of beauty according to Danti.

In 1739, almost 180 years later, Giuseppe Ignazio Rossi would push this tendency to find proportion in the Ricetto even further in his monograph on the Biblioteca, La Libreria Mediceo-Laurentiana, architettura di Michelagnolo Buonarotti, attempting to retrieve a proportional system of the walls and their orders. His conclusion was that Michelangelo here developed a new order, ‘Dorico-Corinthio’, with a ratio of diameter to shaft height of 1 to 5.5, but in which the pedestal and shaft are quite close to the Corinthian order in their detailed articulation. It cannot be established whether Rossi actually measured the Biblioteca, and if so, by what method. In fact, Rossi’s monograph can be read as one big essay in normalization, submitting the building to the uniformizing representational regime of orthogonal and sectional projection, which robs the staircase of its dynamism, and the room as a whole of its play of light and above all shadow achieved by the pietra serena ornament. Indeed, Rossi concludes that the pure force of the underlying proportional system he attributes to the Ricetto makes it perfect, without the need of the ‘external dress of studious ornament’.4 It is an admirable attempt at ignoring or downplaying what cannot be accounted for — the strangeness of the ornament combined with its hidden, but quite compelling logic, which makes all the difference between the genius of Michelangelo and the talent of his followers, by hypothesizing what can be understood in mathematical, that is, rational terms: Michelangelo’s development of a new proportional system.

So up to now we have found several ways of accounting for the strangeness and novelty of the Ricetto: by trying to discover a proportional system, or a new, Dorico-Corinthian order; that is, by relating Michelangelo’s innovations to existent precedents and thereby normalizing them; or, as in the case of Lenzoni, by drawing the Ricetto into the sphere of debates about the Tuscan dialect, and thereby assimilating the mode of utterance and address of the Biblioteca Laurenziana to that of rhetoric. Lenzoni, like Bocchi before him, did so by taking the concept of composto, with its rhetorical associations of compositio — since Alberti, one of the major cases, if not the major case, of using a rhetorical concept to discuss artistic design — as the guiding metaphor to articulate the experience and strangeness of the Ricetto. Danti went even further, endowing the rhetorical concept of composto ordinato with Aristotelian associations because he made it the result of fitness for purpose, or teleological unity of the parts.

To return for a moment to Michelangelo’s drawings for San Lorenzo, with their juxtapositions of architectural and human profiles, we might go somewhat further in the rhetorical analysis of Michelangelo’s overall design or individual ornaments in the Biblioteca, with its anthropomorphic references and analogies, and read it in terms of the rhetorical figure of style called hypotyposis. Derived from the Greek túpos, or type, it referred initially to an image or model impressed on the mind. Subsequently, it came to mean a verbal or visual image, or a series of illustrative details, that makes a strong impression on the mind. Michelangelo’s couplings of human forms and architectural elements can be termed hypotyposes: by their abundance of descriptive detail, they make vivid and imprint on the mind their analogy, or even metaphor, of architectural and human profiles, with their underlying tertium comparationis.

Assumptions of proportional systems in the Ricetto thus served as the basis for various narratives about the room: narratives of composto, ordine, and varietà, of novelty and originality, and fitness for purpose. This analysis of viewers’ reactions to the Biblioteca Laurenziana can be pushed even further. The square dimensions of the Ricetto suggest that there may be easily perceptible relations between dimensions on a very basic level. But as the various attempts to account for its uniqueness discussed here show, there is no evident, consistent or even logical relation between the proportions as observed and the conclusions drawn from them. Rather, it seems as if the proportions viewers think they observe here function as the first guide to making sense of what they see. They have a function in making the viewers think and reformulate what they see in terms of Aristotelian teleology, rhetoric or the questione della lingua, but there is no inherent quality in the observable relations between measurements to connect them in a necessary or sufficient way to these interpretations — let alone to the Ricetto’s beauty. That is, they have a hermeneutic or epistemological role in the process of the viewer’s perception, but they are not inherent aesthetic qualities of the design.

In Michelangelo’s writings or designs there is very little demonstrable interest in the Platonic legitimation of proportional systems that had become current in sixteenth century theology and metaphysics, and adopted late in the century by the more philosophically inclined architectural theorists, such as Daniele Barbaro and Vincenzo Scamozzi (e.g., in Scamozzi 1615: xxiv, 1–3, 5).5 For him mathematical features of the human body, such as symmetry, do not reflect the divine mathematical structure of the universe or of man, but the actual, physical appearance of the human body, as is shown by the letter to an unknown prelate (Buonarroti 1983, 5: 123; Summers 1981: 418–446). A major step in the deconstruction of the Platonist conceptual substructure of proportional systems was made, as is well known, by Claude Perrault in the Ordonnance, with his distinction between positive architectural beauty founded on reasons that convince everybody and arbitrary beauty based on the effects of habits and association; the appreciation of proportion belongs to the latter category (Claude Perrault 1683: i–x). What is perhaps slightly less well known is the way his brother Charles takes this movement even further in the Parallèle des anciens et des modernes of 1688, published five years after the Ordonnance. Here he asked whether the capacity to employ the figures of speech of rhetoric is innate and therefore universal, and compares that capacity to that of using the orders, which according to the defenders of the ancients is also innate. By implication he also questions whether the capacity to use proportional systems is innate, and hence universal and not in need of being taught (Perrault 1692: 126–177). He thus leaves room to entertain the idea that the use of the orders and, by extension, of proportion is a matter of culture and instruction, and hence capable of development or innovation, and liable to decay or even to become obsolete. By this move, proportion became part of the domain of human culture, subject to convention and change, and liable to the threat of relativism.

In a section of Architectural Principles titled ‘The Break-away from the Laws of Harmonic Proportion in Architecture’, Rudolf Wittkower describes 18th-century developments in the use and theory of what he calls proportion, without distinguishing between mathematical proportion and proportion-as-beauty, as a matter of decline and increasing relativism, caused mainly by the impact of British Empiricist aesthetics. By locating the foundation of judgments on beauty, so his argument runs, in the sensuous experience of the subject, and not in the measurable qualities of the object, beauty judgments lose their objective basis, and with this shift proportion lost its foundational role as the basis for beauty as well. This move had been prepared by Claude Perrault’s voicing of social relativism in the Ordonnance, connecting an appreciation of the orders to the social standing of those who propagate them, that is the King and his Académiciens, and his brother Charles’s implicit presentation of cultural relativist views in the Parallèle, when he opens the possibility of conceiving that the orders, like rhetoric, may be a contingent feature of European classical culture.

One of the most explicit cases of the impact of empiricism on architectural aesthetics, however, Julien-David Le Roy’s ‘Essai sur la théorie’ included in the second edition of his Les ruines des plus beaux édifices de la Grèce antique of 1770, points in a different direction from the one Wittkower suggests. Instead of a statement of empiricist aesthetic relativism, the chapter entitled ‘Des Principes de l’Architecture qui dépendent de notre âme, et de notre vue’ is an attempt to understand how architectural beauty can arise once the argument for its objective basis has lost its validity in the interaction between a building, the spectator and the environment (Van Eck 2015). Le Roy does not have in mind a unique individual, endowed with a biography and with a private and personal taste, but the manner in which the perceptual and cognitive make-up of all human beings conditions their perception of buildings. Although Le Roy agreed with the Perrault brothers and John Locke that there is no fixed, inherent, objective basis for beauty, he is no subjectivist. Beauty, we might say using the terminology Locke developed, is not a primary quality, like weight or number, but a secondary one, arising in the mind’s perception of sense impressions (Geraghty 2011: 125–143).

In Le Roy’s analysis, there are first principles, general axioms such as the laws of mechanics, whose application does not result in beauty; and there are secondary principles, less general and secure, that do lead to beauty: ‘the agreeable sensations, the force or the variety of sensations that Architecture makes us feel’.6 Le Roy’s analysis applies in particular to colonnades or peristyles, as his well-known description of viewing the east colonnade of the Louvre (1672) illustrates:

That is, the empiricist shift from an objective beauty inherent in a building and based on a system of mathematical relations between the parts and the whole, to a subjective definition of beauty as a sensation or experience of beauty that arises in the interaction between the building, its environment, its viewer and viewing conditions is not, as Wittkower and James Ackerman argued, a shift from objective rules to individual, subjective feelings and emotions. Instead, Le Roy attempts to demonstrate how the sensation of beauty arises from the fit between the object, its perception, and the human sensory and cognitive apparatus. In this, Le Roy was much influenced by Montesquieu’s Essai sur le goût, published in 1757 and a major step in introducing British empiricist thought in France (Becq 1994: 437–474).7

Also, in Le Roy’s account of viewing the Colonnade the experience is triggered by the placement and proportioning of its columns. It is no longer a narrative of composto ordinato or fitness for purpose of the building. Instead it has become a narrative about form and light, about the changing perception of the colonnade as the light changes or the spectator changes her position. It is, as Sigrid de Jong has argued in her recent Art History article on architecture and theatricality, a plot about the first perception, gradual apprehension, and understanding of a building in the interaction of architecture, situation, and spectator (De Jong 2010: 334–352). But in Le Roy’s case it is also an account of almost empty, purely sensual narrative, a proto-phenomenological experience of architecture, light and shade, solids and voids, columns and arcades. It is a recollection of an aesthetic experience in the 18th-century sense, prefiguring Kant’s definition of the experience of beauty as the harmony of a human being’s powers of perception and cognition. Le Roy’s narrative also reconstitutes the awareness and articulation of a physical experience of inanimate stone which to the viewer appears to become animated by the changes caused by the trajectory of the sun and the movement of the spectator’s gaze (Le Roy 1770: vii). Le Roy’s narrative of viewing the colonnade documents the change, in 18th-century architectural aesthetics, from the attribution of proportion to a building as a way of accounting for its beauty, to a new, subjective experience of beauty in which empathy, or Einfühlung — endowing lifeless stones with human emotions and memories of physical experiences — takes over from a more analytic attitude.

Now let us return for a moment to Michelangelo’s drawings of a human profile merged with an architectural profile (Figs. 1, 2). It is unclear what these images purport to say: that the human face has a formal architectural substructure? that the proportions of architecture can best be understood, or figured out, as human forms? that there is a fundamental similarity between the forms of the human face and body and those of pediments or architraves? I also mentioned that these sheets may be interpreted, drawing on rhetoric, as cases of hypotyposis: as a wealth of visual, illustrative detail that allow the images to be forcibly impressed on the mind of the viewer. Leaving the historical case of Michelangelo’s Ricetto and the attempts of early modern viewers to make sense of it, we touch here on a fundamental question in human cognition: how does the human mind make sense of, articulate or conceptualize sense perception? Can we say that perceptions of buildings are influenced by geometrical or numerical proportion, or that they resemble human faces? Does it still make sense to argue that the best conceptualization of architecture is in terms of mathematics or anthropomorphy?

Kant addressed this problem in the Critique of Judgment, published in 1790, which combines his critique of aesthetic judgment with that of teleological judgment. I will not go into detail here, but just want to discuss briefly section 59, where he defines what he calls Schematismus: to decide, by an act of judgment, how to define sense perceptions. To illustrate this cognitive process he calls on the rhetorical concept of hypotyposis, describing it as Versinnlichung, making an abstract concept visual or perceptible by clothing it, filling it, with sensuous experience.

Architecture plays an interesting role in Kant’s thought. He rarely wrote about it, with a few notable examples such as citing Saint Peter’s as a case of the sublime in his early essay on the sublime (Kant 1905–1912: 239). His late work, however, is full of architectural metaphors, which structure his presentation of his system in the three Critiques. Schematismus as used in the Critique of Judgment is also a metaphor with strong architectural connotations: the filling in of a structure with material and ornament.8

In this context he does not mention proportion or anthropomorphy, but I think we are justified in presenting Michelangelo’s drawings, and the attempts we have discussed so far to make sense of the perception of the Ricetto, as instances of Kantian schematism, of hypotyposis in Kant’s adaptation of a rhetorical strategy into a way of understanding what goes on in the mind when people try to make sense of their perceptions.

But there is also another way of connecting Kant with the reactions to the Biblioteca, and in particular Danti’s presentation of proportion as the visual manifestation of a body’s fitness for purpose. The main type of schematism Kant discusses in the Critique of Judgment is that of teleology or Zweckmäßigkeit. According to him the appearance and structure of natural organisms and their parts — eyes, human bodies, the claws of birds, and the fangs of wolves — can only be fully understood if we consider them not as contingent aggregates of matter, but as if designed by God or nature, in the neo-Platonic sense of natura naturans, with their purpose in mind: their structure is one of fitness for purpose. In the case of natural organisms there is no such prime creator, according to Kant, but to understand them, the human mind uses the concept, or schema in his terminology, of teleology or fitness for purpose. It can only grasp why the human eye looks the way it does, its shape and organization, if it knows that its purpose is to function as the organ of visual perception. In the case of art, however, there is the artist, the prime mover or auctor intellectualis, who has conceived the work of art with its end in mind.

It is this Aristotelian, objective fitness for purpose which Kant takes over, but changes into a concept, a hypothesis to make sense of human perceptions of the natural world and art. I would indeed argue that according to the Critique of Judgment there are two fundamental groups of schematisms: a mathematical one and an organic one.9 Peter Kidson’s statement that the medieval use of, and search for, proportional systems is an expression of the first group: the belief that all forms can ultimately be expressed in number illustrates such mathematical schematism (Kidson 1997). The very widespread human tendency to speak of works of art as if they were living beings may be considered a representation of the second group. The use by Pevsner, Ackerman, and Brothers of empathetic reactions to the Ricetto mentioned at the opening of this paper are also examples of the second group, in that they all attribute characteristics of human life and emotions to stone forms. And again, we might return to Michelangelo’s drawings as suggestions of this double way of trying to understand perception: in either mathematical or organic, that is, anthropomorphic, terms.

One of the very first essays Wittkower wrote, his ‘Das Problem der Bewegung innerhalb der manieristischen Architektur’ of 1933, bears on the issues presented here (see author’s translation in the Appendix).10 It is about movement in Mannerist architecture, and discusses among other works of architecture Giuliano da Sangallo’s Gondi Chapel in Santa Maria Novella of 1503–1506, a model for the Ricetto, as Cammy Brothers has recently argued (Fig. 3; Brothers 2008: 191).

Of the triumphal arch motif framing the altar, Wittkower wrote that each pilaster carries its own architrave, but that it is unclear to which bay, the outer or the inner, the inner pilasters belong. This ambiguity is strengthened by the egg-and-dart list running unbrokenly above them. The tectonic ambiguity of these pilasters creates a feeling of movement — not in the building, but in the viewer, whose gaze cannot settle. When the viewer’s eye follows one set of directions it is confronted with a contradictory set. Hence, Wittkower concludes in an ambiguous formulation which is typical of the aesthetics of empathy, ‘a moment of movement is inherent in the architecture’. This unsettled conflict of directions leads to ‘an oscillating emotional state’, characteristic of unsettled times such as the 1530s and 1540s ‘with their desperate need for clear solutions’.11

It is totally unlike Wittkower’s post-war work. Like Le Roy’s description of the colonnade of the Louvre, it is an attempt to put into words the experience of architecture by drawing not on the scripts or narratives of proportion or the orders, but on the physical experience of architecture as movement and animation. It is, in fact, an exercise in the aesthetics of Einfühlung, of the animation of lifeless objects by endowing them with the emotions and sensations of the spectator, about which Wittkower’s first teacher, Wölfflin, had written so eloquently in his Prolegomena to a Psychology of Architecture:

After the war, Wittkower would react completely against this approach, both against the aesthetics of Einfühlung and the notion of pure form. But we can see, both in the 1933 essay and in Architectural Principles, two ways, albeit very different ones, of coming to terms with the epistemological challenge classical architecture in general, because of its absence of iconic elements, and proportion in particular, because of its abstract and general character, pose to the viewer: how to make sense of, how to articulate, visual perception. And interestingly enough, Wittkower draws on both large categories of Schematismus that Kant had distinguished: in the 1933 essay about movement in architecture, he draws on anthropomorphy and empathy; in his mature Architectural Principles, on the mathematical approach of proportion analysis. But in both cases, he, as did so many other thinkers about proportion, tried to make sense in artistic terms of what is ultimately an epistemological problem: how to understand sense perception.

The argument I have presented here starts from the ambiguity presented by Michelangelo’s sketch. Is he thinking of the human face in terms of architectural structure, or of architectural form in terms of the human face? The reactions by early modern viewers to the puzzling singularity of the Ricetto illustrate two ways in which all human viewers try to make sense of what they see: by assuming proportional systems, even if there is no clear, objective indication for them; or by assimilating architecture with other forms of human communication, in particular language and rhetoric. The second tendency would lead to the aesthetics of empathy, which is still very present in recent comments by the historians Pevsner, Ackerman, and Brothers. In the second part of my paper, moving from the historical reactions to aesthetic and epistemological aspects, I argue that, contrary to the view proposed by Wittkower in Architectural Principles, the impact of Empiricist thought on proportion theory, or architectural theory in general, did not result in an increasing subjectivization of architectural beauty. Instead, as shown in Le Roy’s account of viewing the Colonnade of the Louvre, it led to a change in how architectural beauty was conceived: not as an inherent, objective quality of the building, of which proportional systems are one of the chief manifestations, but as a sensation of beauty resulting from the fit between the object perceived and the human mind. Kant would take this argument much further. In his Critique of Judgment he argued how the human mind needs Schematismus, filling in concepts with sense perception, to make sense of perception, and that two major assumptions are needed for the mind to achieve this: that of teleology and that of mathematical structure. Kant’s argument thus offers the most radical answer to the question raised in so many of the essays in this volume: why do viewers attribute proportional systems to buildings. They do so not because of an inherent mathematical quality of the building, but because it helps them to make sense, to put into words and conceptualize, what they see. Ultimately proportional systems are constructions of the mind.