This work aim to analyse relations between Arithmetic and Geometry indicated by Piero della Francesca in his treatise Libellus de quinque corporibus regularibus. Piero della Francesca was a painter and scholar of perspective, geometry and arithmetic, in his time. He carried out investigations on pictorial, geometric and architectural issues. Of the treatises he wrote, only three are preserved, on perspective, Geometry and Arithmetic. The central document selected for this research was the manuscript Libellus de quinque corporibus regularibus, (...) deposited in the Vatican Apostolic Library in digitized copy. Throughout this work we have tried to understand the Libellus de quinque corporibus regularibus as a result of the work of an artist and scholar who proposed ways of relating Arithmetic and Geometry using both his practical knowledge and the works of authors of antiquity. (shrink)

The main conclusion of Nigel Hiscock's important but ill‐arranged book is that the ground plans of abbeys and cathedrals of the tenth and eleventh centuries incorporate Platonic wisdom—hence the “wise” in the title catchwords, which come from Paul's first letter to the Corinthians . There Paul likens himself to a sapiens architectus who lays the foundations on which others erect the building. In three of the four translations in The Complete Parallel Bible, however, Paul does not declare himself wise but, (...) rather, “skilled” or “trained.” The discrepancy makes an emblem for Hiscock's investigation: Did the architects deliberately express geometrical ratios significant in Platonic philosophy in their designs or did they proceed by applying their practical skill and training to the lie of the land and the resources available? A subsidiary question is also symbolized in Paul's verse: Was the designer a wise cleric who set the proportions and left the heavy work to someone else?Hiscock builds his answers in two ways. For one, he delivers a chrestomathy of Platonic snippets known in the tenth century and a few telling quotations from wise clerics—the latter concern the expression of significant numbers in the church fabric: a baptistery is octagonal because of an association between creation and the number eight, and a church with a half‐dozen altars is complete because of the perfection of the number six. But, Hiscock emphasizes, numerology does not account for floor plans, and to test the wisdom of the designers he scrutinizes plans of two dozen churches, especially those associated with monastic reformers. He expresses his results in color‐coded lines superposed on the plans, some one hundred of which comprise a valuable appendix.An example will indicate the interest of the findings and the riskiness of the business. The crossing of the transept at St. Michael's Abbey, Hildesheim, makes a square ABCD, proceeding clockwise from A in the northeast corner. AD and BC prolonged give the line of nave piers, AB and DC prolonged that of the arms of the transept. Lines drawn from B at angles of 54° and 60° with BC cut CD extended in E and F, respectively, locating the end piers in the side aisles. Now 60°, as the invariable angle of a three‐sided figure all of whose parts are equal, carries a cornucopia of Platonic and Christian symbolism. Also 54°, because derivable from a pentagon, has the pregnant significance of the figure five, which is the sum of the first male and first female numbers. Repetition of the transept square fixes the nave piers. Then lines between piers, pillars, and pilasters make the special angles with the north‐south and east‐west lines of the grid all over the plan.How much of this construction is mere coincidence? Requiring the intervals between piers in the same line to be equal, we have DE = FE = FD/2. And so it is: / = 0.51. Otherwise put, if angle FBC is 60° and E divides FD equally, angle EBC must be 54° without reference to a pentagon. Against such arguments, Hiscock observes that his analysis has uncovered previously unknown relations between churches confirmed by documents and that his technique, when applied to large nineteenth‐century buildings with repeated bays such as the Crystal Palace, picks out relatively few structurally significant elements.Hiscock is senior lecturer on the design, theory, and history of architecture at Oxford Brookes University and a master craftsman. His scrupulously prepared color‐coded drawings contain much of interest and pleasure for a geometer. Has he definitively answered the old vexed question whether the sapiens architectus intended to express ancient wisdom in his plans? If the architect thus proposed, did his builder faithfully dispose? Paul again may give guidance. “The Lord knows the thoughts of the wise, that they are futile”. (shrink)

During his stay in Padua ca. 1592–1610, Galileo Galilei (1564–1642) was a lecturer of mathematics at the University of Padua and a tutor to private students of military architecture and fortifications. He carried out these activities at the Academia degli Artisti. At the same time, and in relation to his teaching activities, he began to study the equilibrium of bodies and strength of materials, later better structured and completed in his Dialogues Concerning Two New Sciences of 1638. This paper (...) examines important details of four works dating to the Paduan period: Breve instruzione dell’architettura militare; Trattato di Fortificazione; Le Mecaniche; Le operazioni del compasso geometrico et militare. The two works on military architecture and fortifications were compiled from notes taken by students, and are not by Galileo’s hand, but are still illustrative of his work and thinking at the time. (shrink)

With In Focus Sacred Geometry, learn the fascinating history behind this ancient tradition as well as how to decipher the geometrical symbols, formulas, and patterns based on mathematical patterns. People have searched for the meaning behind mathematical patterns for thousands of years. At its core, sacred geometry seeks to find the universal patterns that are found and applied to the objects surrounding us, such as the designs found in temples, churches, mosques, monuments, art, architecture, and nature. Learn (...) the fundamental principles behind: Interpreting the sacred symbolism and principles behind shapes Calculating numbers used in sacred geometry Applying the golden ratio and Fibonacci’s Sequence to objects Translating symbolic and geometrical letters and alphabets This accessible and beautifully designed guide to sacred geometry includes a frameable poster of the main sacred geometrical shapes and their unique, innate arrangements. The In Focus series applies a modern approach to teaching the classic body, mind, and spirit subjects. Authored by experts in their respective fields, these beginner's guides feature smartly designed visual material that clearly illustrates key topics within each subject. As a bonus, each book includes reference cards or a poster, held in an envelope inside the back cover, that give you a quick, go-to guide containing the most important information on the subject. (shrink)

Neuroarchitecture uses neuroscientific tools to better understand architectural design and its impact on human perception and subjective experience. The form or shape of the built environment is fundamental to architectural design, but not many studies have shown the impact of different forms on the inhabitants’ emotions. This study investigated the neurophysiological correlates of different interior forms on the perceivers’ affective state and the accompanying brain activity. To understand the impact of naturalistic three-dimensional (3D) architectural forms, it is essential to perceive (...) forms from different perspectives. We computed clusters of form features extracted from pictures of residential interiors and constructed exemplary 3D room models based on and representing different formal clusters. To investigate human brain activity during three-dimensional perception of architectural spaces, we used a mobile brain/body imaging (MoBI) approach recording the electroencephalogram (EEG) of participants while they naturally walk through different interior forms in virtual reality (VR). The results revealed a strong impact of curvature geometries on activity in the anterior cingulate cortex (ACC). Theta band activity in ACC correlated with specific feature types (rs (14) = 0.525, p = 0.037) and geometry (rs (14) = - 0.579, p = 0.019), providing evidence for a role of this structure in processing architectural features beyond their emotional impact. The posterior cingulate cortex and the occipital lobe were involved in the perception of different room perspectives during the stroll through the rooms. This study sheds new light on the use of mobile EEG and VR in architectural studies and provides the opportunity to study human brain dynamics in participants that actively explore and realistically experience architectural spaces. (shrink)

In Résolution des quatre principaux problèmes d’architecture (1673) then in Cours d’architecture (1683), the architect–mathematician Nicolas-François Blondel addresses one of the most famous architectural problems of all times, that of the reduction in columns (entasis). The interest of the text lies in the variety of subjects that are linked to this issue. (1) The text is a response to the challenge launched by Curabelle in 1664 under the name Étrenne à tous les architectes; (2) Blondel mathematicizes the problem (...) in the “style of the Ancients”; (3) The problem is reformulated and solved through the continuous drawing of the curve; (4) Blondel refutes the uniqueness of the curve by enumerating a variety of solutions (conchoid, spiral, parabola, ellipse, circle, hyperbola). This exuberance responds to an intention that does not coincide with the state of the art of mathematics at the end of the seventeenth century, nor with the taste for geometry of the Ancients, nor with any pedagogical project. This feature is explained by Blondel’s plan to found architecture on scientific bases. The reasons for his failure are analysed. (shrink)

This PhD responds to a two fold problem with the philosophy of design and the practice of design. The philosophical problem is stated as the discrepancy between a dominant philosophical framework that orders the world according to eternal essences and the actual conditions of the world in which we exist: the conditions of becoming and of flux. Commencing with a critique of the western metaphysical tradition of statics this research project proposes that we need to find a way of describing (...) an evolutionary model of practice, and by so doing to provide a revitalised narration of process. Consequently, the PhD explores the meanings of process through a critical examination of an ensemble of projects created by the author. Within this framework, a number of questions are posed in order to explore the proposition of a process practice. These questions are: What is process? How does one think process? Indeed, how do we get to grasp change? What are the consequences of process thinking on the practices of design, their fields of operation, and their productions? And, how can the thematising of process contribute to the design of the constructed environment, as well as reconfigure the practices of design? This thematising of process is argued to involve a necessary address to the constitutive and interrelated characteristics of process: space/time, movement, change, form and matter. Such an address is also seen to problematise the status of the object, the paradigms of representation, the modes of creation, the economies of exchange, and the structures of community, and to offer a modality of practice which would re-imagine the forms of social exchange to offer an ethical alternative to the tyranny of supply and demand, and thereby reconfigure the potential for dwelling. Making an overview of the discourses and practices engaging with theories of becoming, this thesis argues that almost all of these re-inscribe statics and that consequently the practice of design seems to drag behind our understanding of the world. Through a meditation on dis/appearance, in which the dynamics of being and becoming and the restless ambiguity of the gap are examined, the work establishes a process vocabulary, and makes clear through a material practice, the domains of process thinking, its inclinations, and the kinds of operations and procedures that flourish there. Foregrounding the fertile character of process practice, the PhD then proceeds to introduce notions of the movement-form, the duration-form, the transformational-form, the geometry of encounter, and to argue for physical form as an in-movement poise. Advocating new modes of approach and of attentiveness, and demonstrating new generative methods, this PhD argues that process thinking is not simply an operational stance, but an ethical position that identifies a field of care, and that consequently the design practices be expanded by taking seriously the relationship between process thinking and place making. Thus, this thesis concludes by advocating a mode of place making which, rather than reproduce planned environments as systems of control, configures place as the discursive contested place of encounter and exchange. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 44.2 (2006) 321-322 [Access article in PDF] Thomas Holden, The Architecture of Matter: Galileo to Kant. Oxford: Clarendon Press, 2004. Pp. x + 305. Cloth $74.00. Most scholars believe that the problem of infinite divisibility that plagued early modern natural philosophy was an entirely mathematical issue and, therefore, resulted from the short-comings of early modern mathematics. Accordingly, advances in geometry, topology (...) and the logic of infinity are often thought to lay this debate to rest, thereby reducing it to some quaint footnote in the history of natural philosophy. Thomas Holden, however, through thorough textual and philosophical analysis, convincingly argues that this debate was more metaphysical in nature than previously supposed. He shows how the various paradoxes concerning the structure of matter stem from two widely-accepted theses central to the new science. The first resulted from the early modern method of applying the principles of geometry to physics. This geometrical method dictated that bodies are metaphysically divisible (m-divisible) in infinitum, because they occupy regions of space that can, at least by the power of God, be subdivided into those parts occupying corresponding sub-regions of space, and those sub-parts into those sub-sub-parts occupying their respective corresponding sub-sub-regions of space in infinitum. The other thesis is that "the parts of material bodies are each concrete existents in their own right" (3). Such actual parts exist prior to any act of division, whereas parts that are only potential would not exist prior to division but only after a divisive act has occurred. Holden argues that the actual parts doctrine was a metaphysical thesis just as fundamental to Enlightenment natural philosophers as the geometrization of matter. Indeed, it is these two theses that came into conflict and gave rise to a plethora of antimonies at the heart of the new science.Holden should be applauded for the skill he employs in challenging the received scholarship. In chapter 1, after some historical and terminological preliminaries, Holden reconstructs several of the more popular paradoxes of material structure found throughout the period's vast literature and successfully shows how these paradoxes would not arise if the problems of material structure were merely mathematical. He shows how the metaphysical doctrine of actual parts gave rise to these difficulties because of its conflict with the assumed geometrical doctrine that extension is infinitely m-divisible. For example, some thought it followed from the actual parts doctrine that there must be an actual infinity of parts, because actual parts exist prior to any process of division and, therefore, in order for both actual parts and infinite m-divisibility to be true, there must be an actual infinity of parts. This raised issues about the intelligibility and piety of an actual infinite as well as issues about equal and unequal infinities, e.g., half an infinite number of parts would still be infinite but half as much as the whole collection of parts. Hence, this and other paradoxes would not arise in the absence of the actual parts doctrine, and so the debate is more metaphysical in nature than previously supposed. [End Page 321]Chapter 2 examines in more detail the actual parts metaphysic and how it leads to tension with infinite m-divisibility. Chapter 3 examines various "short-circuit" arguments designed to undermine the actual parts doctrine. And chapter 4 examines the actual parts doctrine via the argument from composition found in Kant's pre-critical work, Physical Monadology. In chapter 5, Holden briefly examines the potential parts doctrine and how some actual-parts partisans tried to undermine it. It is eventually concluded that actual-parts and potential-parts advocates are at an impasse due to the question-begging nature of their respective arguments. Finally, chapter 6 is "something of an addendum" (74), since it examines a fascinating synthesis of the two conflicting positions, viz. the Kant-Boscovich force-shell atom theory.Given that the book is about material structure, it is a propos that my critical comment is about the structure of the book. It seems... (shrink)

The overarching guiding principle of Alan Turing’s work was directed towards modelling the human mind as a machine. It is extraordinary that Turing introduced, in his early papers, ideas that are only now beginning to be investigated. Throughout his life, he considered conjectures to be of great importance because they suggest useful lines of research. In my own conjecture, I am asking the question: what is the brain’s geometry? Can it ever be unravelled, or does its complexity defy any (...) form of visual representation? Recent research reveals that the detailed workings of the brain take place in a ‘global workspace’, where conscious contents appear to be disseminated globally to a great multitude of networks throughout the brain that are unconscious. During conscious tasks, neurons contributing to the global workspace enter into a coherent activation pattern by being tightly connected through long axons. Is it possible, therefore, that my proposed geometric pattern (an Islamic latticework) can be envisaged as a useful means of interconnecting multiple long axons in a unified global workspace dedicated to conscious thought? I share and communicate my idea through a process that combines visual imagery with musical performance (i.e. visual music). It is a method that can assist in clarifying concepts that might otherwise remain elusive or esoteric. (shrink)

How we experience space by listening: the concepts of aural architecture, with examples ranging from Gothic cathedrals to surround sound home theater. We experience spaces not only by seeing but also by listening. We can navigate a room in the dark, and "hear" the emptiness of a house without furniture. Our experience of music in a concert hall depends on whether we sit in the front row or under the balcony. The unique acoustics of religious spaces acquire symbolic meaning. (...) Social relationships are strongly influenced by the way that space changes sound. In Spaces Speak, Are You Listening?, Barry Blesser and Linda-Ruth Salter examine auditory spatial awareness: experiencing space by attentive listening. Every environment has an aural architecture.The audible attributes of physical space have always contributed to the fabric of human culture, as demonstrated by prehistoric multimedia cave paintings, classical Greek open-air theaters, Gothic cathedrals, acoustic geography of French villages, modern music reproduction, and virtual spaces in home theaters. Auditory spatial awareness is a prism that reveals a culture's attitudes toward hearing and space. Some listeners can learn to "see" objects with their ears, but even without training, we can all hear spatial geometry such as an open door or low ceiling. Integrating contributions from a wide range of disciplines—including architecture, music, acoustics, evolution, anthropology, cognitive psychology, audio engineering, and many others—Spaces Speak, Are You Listening? establishes the concepts and language of aural architecture. These concepts provide an interdisciplinary guide for anyone interested in gaining a better understanding of how space enhances our well-being. Aural architecture is not the exclusive domain of specialists. Accidentally or intentionally, we all function as aural architects. (shrink)

Contemporary architecture is increasingly grounded in science and mathematics. Architectural discourse has shifted radically from the sometimes disorienting Derridean deconstruction, to engaging scientific terms such as fractals, chaos, complexity, nonlinearity, and evolving systems. That's where the architectural action is -- at least for cutting-edge architects and thinkers -- and every practicing architect and student needs to become conversant with these terms and know what they mean. Unfortunately, the vast majority of architecture faculty are unprepared to explain them to (...) students, not having had a scientific education themselves. Here is an architecture book by an architect/scientist, just in time to help architects in the new millennium. Alexander discusses many of the scientific terms arising in cutting-edge architecture, and explains them to those who don't have scientific training or advanced mathematical knowledge. We find discussions of the evolution of forms; the importance of process in design; iteration; genetic algorithms; sequences of transformations; different levels of scale (i.e. fractals); etc. They are explained here by an architect who is also a scientist, because he wants to change the way architects think and build. Alexander is not merely popularizing other scientists' results and making them accessible to architects: he is in fact presenting new and original scientific work that ties many of these concepts together in a way that will be useful to architects. (shrink)

Using as a starting point recent and apparently incompatible conclusions by Saunders and Knox, I revisit the question of the correct spacetime setting for Newtonian physics. I argue that understood correctly, these two versions of Newtonian physics make the same claims both about the background geometry required to define the theory, and about the inertial structure of the theory. In doing so I illustrate and explore in detail the view—espoused by Knox, and also by Brown —that inertial structure is (...) defined by the dynamics governing subsystems of a larger system. This clarifies some interesting features of Newtonian physics, notably the distinction between using the theory to model subsystems of a larger whole and using it to model complete universes, and the scale-relativity of spacetime structure. _1_ Introduction _2_ Newtonian Mechanics and Galilean Spacetime _3_ Vector Relationism and Maxwellian Spacetime _4_ Recovering the Galilei Group: Dynamics of Subsystems _5_ Knox on Inertial Structure _6_ Connections on Maxwellian Spacetime _7_ Knox on Newtonian Gravity _8_ Vector Relationism and Newton–Cartan Theory _9_ Inertial Structure in Newton–Cartan Gravity _10_ Reconciling Knox and Saunders _11_ Conclusions. (shrink)

According to Kepler and Descartes, the geometry of the triangle formed by the two eyes when focused on a single point affords perception of the distance to that point. Kepler characterized the processes involved as associative learning. Descartes described the processes as a “ natural geometry.” Many interpreters have Descartes holding that perceivers calculate the distance to the focal point using angle-side-angle, calculations that are reduced to unnoticed mental habits in adult vision. This article offers a purely psychophysiological (...) interpretation of Descartes’s natural geometry. In his account of perceived limb position from the Treatise on Man, he envisioned a central brain state that controls ocular convergence and thereby co-varies with the distance from observer to object. A psychophysiological law relates the visual perception of distance to this brain state. Descartes also invokes more traditional theories of distance and size perception based on unnoticed judgments, yielding a hybrid account. (shrink)

Customary interpretations state that Tractarian thoughts are pictures, and, a fortiori, facts. I argue that important difficulties are unavoidable if we assume this standard view, and I propose a reading of the concept taking advantage of an analogy that Wittgenstein introduces, namely, the analogy between thoughts and projective geometry. I claim that thoughts should be understood neither as pictures nor as facts, but as acts of geometric projection in logical space. The interpretation I propose thus removes the root of (...) the identified difficulties. Moreover, it allows important clarification concerning some central aspects of the Tractarian theory of representation, and it yields a unifying elucidation regarding Wittgenstein’s remarks on the solipsistic thesis. (shrink)

The paper presents the diagrammatic calculus CL, which combines features of tree, Euler-type, Venn-type diagrams and squares of opposition. In its basic form, `CL' (= Cubus Logicus) organizes terms in the form of a square or cube. By applying the arrows of the square of opposition to CL, judgments and inferences can be displayed. Thus CL offers on the one hand an intuitive method to display ontologies and on the other hand a diagrammatic tool to check inferences. The paper focuses (...) mainly on the adaptation of the square of opposition in CL and offers an algebraic notation, which corresponds to the diagrammatic representation. (shrink)

It may seem reasonable, even inevitable, that architectural practice should be based on an understanding that architects, like lawyers and doctors, should discover their clients' needs and accommodate them to the best of their abilities. But current discussion within the legal and medical professions of the conflict between service to private individuals who can pay, and to the public who cannot, suggest an expanded or altered definition of professional responsibility. Actually, the conflict between public and private interest may be more (...) acute in architecture than in other professions: the kind of buildings architects design are costly and are made possible only by the wealth of a small segment of the population or the state, yet every one raised affects the lives of people other than the one who makes the program and pays the architect for his services. Furthermore, the decisions of architects are embodied in buildings that last for generations, even for millennia, so that the overwhelming majority of people in our culture live and work in places designed not only for other people but for other times and conditions. For this reason, even the "private" practice of architecture involves responsibilities to a widespread constituency. James S. Ackerman is the author of The Architecture of Michelango, Art and Archaeology, The Cortille del Belvedere, Palladio and Palladio's Villas and is professor of fine arts at Harvard University. He has contributed "On Judging Art without Absolutes" to Critical Inquiry. (shrink)

This paper examines Hobbes’s criticisms of Robert Boyle’s air-pump experiments in light of Hobbes’s account in _De Corpore_ and _De Homine_ of the relationship of natural philosophy to geometry. I argue that Hobbes’s criticisms rely upon his understanding of what counts as “true physics.” Instead of seeing Hobbes as defending natural philosophy as “a causal enterprise … [that] as such, secured total and irrevocable assent,” 1 I argue that, in his disagreement with Boyle, Hobbes relied upon his understanding of (...) natural philosophy as a mixed mathematical science. In a mixed mathematical science one can mix facts from experience with causal principles borrowed from geometry. Hobbes’s harsh criticisms of Boyle’s philosophy, especially in the _Dialogus Physicus, sive De natura aeris_, should thus be understood as Hobbes advancing his view of the proper relationship of natural philosophy to geometry in terms of mixing principles from geometry with facts from experience. Understood in this light, Hobbes need not be taken to reject or diminish the importance of experiment/experience; nor should Hobbes’s criticisms in _Dialogus Physicus_ be understood as rejecting experimenting as ignoble and not befitting a philosopher. Instead, Hobbes’s viewpoint is that experiment/experience must be understood within its proper place – it establishes the ‘that’ for a mixed mathematical science explanation. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:…duplici modo Daemon homini carnaliter copulatur:Ludovico Maria Sinistrari's Alternative to Apostasy and Sorcery in Human-Incubus IntercourseBert RoestLodovico Maria Sinistrari d'Ameno (1632-1701), who joined the Riformati branch in 1647 in the Pavian Provincia di S. Diego, is one of the many productive seventeenth-century Franciscan authors whose works are not habitually discussed within the world of Franciscan scholarship. According to the existing bibliographical guides, Sinistrari authored under his own name and (...) under various pseudonyms (such as Panfilo, Clodoveo Farvamondi, Nicolò Turris, Lazaro Socio, and Lazaro Agostino Cotta) about 34 works, more or less half of which reached the printing press during his lifetime. These works cover a wide range of genres and topics, including comedies, religious and secular dramas, astrological, astronomical and scientific works (including polemics against works of other scholars, medical and embryological papers), defenses of the religious orders, sermons and funerary orations, treatises of sacramental theology, some atypical hagiographical texts, and works of religious disciplinary law. This production was linked to his acknowledged expertise in a variety of disciplines, which apparently made him a celebrated lecturer in Pavia within the order and in public schools, teaching the liberal arts, geometry, military architecture and theology. Some of his works are connected with his activities as a preacher, advisor, and later censor for the Holy Office, practicing exorcist, consultant for the Franciscan Minister General, and personal theologian of Alessandro Montecatini (Archbishop of Avignon) and Federico Caccia (Cardinal and Archbishop of Milan).1Within the order itself, his most enduring legacy might have been his three volume Practica criminalis illustrata, a work that is both a commentary on penal law within the Franciscan order, and a more wide-ranging manual for dealing with penitential and religious-criminal issues, written [End Page 191] to supplant older handbooks of such kind. He was initially tasked to compile such materials at the Franciscan general chapter celebrated at Rome in 1688. The first two volumes of the Practica criminalis illustrata appeared in print in 1693. They were again published posthumously in 1702.2 In between he finished another volume of the Practica with a somewhat different focus, also known under the title De delictis et poenis tractatus absolutissimus, and which was proposed as a useful guidebook for judges, lawyers, and prelates in ecclesiastical courts, as well as for confessors, exorcists, and lay practitioners of law. This work was issued for the first time in 1700.3Although a wished-for early eighteenth-century re-issue of this multi-volume Practica criminalis illustrata was put on hold when it was placed on the index of forbidden books in March 1709 donec corrigatur for suspicion of laxist tendencies,4 all three volumes did appear again with some modifications in a 1753-1754 opera omnia edition. It could be found in a significant number of Franciscan and non-Franciscan order libraries, and apparently remained for a long time an important resource for penitential and disciplinary policies.5 [End Page 192]Whereas Sinistrari, like so many early modern friars, is not a commanding presence in the scholarly and literary canon of present-day Franciscan order history, he became a figure of some fascination and even notoriety in nineteenth- and early twentieth-century esoteric circles, and also is known by modern scholars of early modern demonology and early modern sexuality. This is mainly due to the publication, in the later nineteenth century, of two of Sinistrari's works on demoniality and sodomy that as such were not printed in full during our friar's lifetime, but parts of which are included in Titulus 4 of his Latin De delictis et poenis tractatus absolutissimus, namely the crimes against chastity (which in De delictis et poenis, alongside of daemonialitas [which comes last], comprises constortium suspectum, sollecitatio ad turpia, fornicatio, stuprum, incestus, raptus, clausurae violatio, sacrilegium cum moniali, mollities, sodomia, and bestialitas).Hence, in 1875, the French book collector and editor Isidore Liseux issued for the first time De la démonialité et des animaux incubes et succubes où l'on prouve qu'il existe sur terre des créatures raisonnables autres que l'homme, ayant comme lui un corps et une âme, naissant et mourant comme... (shrink)

One of the important modernist terms – flexibility – offers the introduction of time and of the unknown as parameters in design. Yet the development of flexibility in modern architecture shows also the ambivalent relationship between architecture and the user – the objective of incorporating flexibility in archi-tecture. The span between introducing the freedom of choice and expression and the reality of totally controlled spaces and movements shows the range of interpretations of this subject. By looking at the (...) strategies and evolution of flexibility in architecture one can project possible problems and frictions for the new discipline – ubiquitous computing. Both flexible architecture and the ubiquitous house are devoted to space, time and technology, and intend to give the user the maximum of possibilities inside the house. What are the mistakes that some of the development in flexible architecture has made, that ubiquitous computing might learn from, especially in its relation to the user? (shrink)

Does geometry constitues a core set of intuitions present in all humans, regarless of their language or schooling ? We used two non verbal tests to probe the conceptual primitives of geometry in the Munduruku, an isolated Amazonian indigene group. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

This paper examines the discrepancy between the attitudes of many historians of mathematics to sixteenth-century geometry and those of museum curators and others interested in practical mathematics and in instruments. It argues for the need to treat past mathematical practice, not in relation to timeless criteria of mathematical worth, but according to the agenda of the period. Three examples of geometrical activity are used to illustrate this, and two particular contexts are presented in which mathematical practice localised in the (...) sixteenth century takes on a special historical significance. (shrink)

Buildings are frequently described as ambiguous and, indeed, they often involve the ambivalence associated with ambiguous symbols. In this paper, I develop a theory of architectural ambiguity within the framework of a Goodmanian symbol theory. Based upon Israel Scheffler’s study of verbal and pictorial ambiguity, I present a theory of denotational ambiguity of buildings which distinguishes four types of ambiguity: elementary ambiguity, interpretation-ambiguity, multiple meaning and metaphor, which proves to be a special case of multiple meaning. Denotationally ambiguous buildings are (...) exceptions, because buildings usually exemplify rather than denote. I therefore add a theory of exemplificational ambiguity. The crucial distinction between mere multiple exemplification and genuine exemplificational ambiguity leads to two versions of each of the first three types of ambiguity. The resulting extension of Goodman’s symbol theory is of interest beyond architecture. (shrink)

In this paper we examine the contributions of the Italian geometrical school to the Foundations of Projective Geometry. Starting from De Paolis' work we discuss some papers by Segre, Peano, Veronese, Fano and Pieri. In particular we try to show how a totally abstract and general point of view was clearly adopted by the Italian scholars many years before the publication of Hilbert's Grundlagen.We are particularly interested in the interrelations between the Italian and the German schools (mainly the influence (...) of Staudt's and Klein's works). We try also to understand the reason of the steady decline of the Italian school during the twentieth century. (shrink)

Examines the ancient cosmic science of the female megalithic astronomers.

Architecture and urbanism seem to be »weak« disciplines, constantly struggling for a better understanding of their nature and disciplinary borders. The huge amount of metaphors appearing in the discourse of both not only reference to their creative nature but also indicate their weakness and the missing piece strengthening their own understanding: a definition of space for architecture and of city for urbanism. But using metaphors in this field implies a problem - though metaphors achieve to bring opposites together, (...) there remains the question how literal they can actually become in order to relate to these subjects properly. In this volume, several authors from various fields using different approaches discuss this question. (shrink)

The article examines the reorientations of the appreciation of ugliness within different national contexts in a comparative and relational frame, juxtaposing the Australian, American, British and Italian milieus. It also explores the ways in which the transformation of the urban fabric and the effect of suburbanization were perceived in the aforementioned national contexts. Special attention is paid to the production and dissemination of how the city’s uglification was conceptualized between the 1950s and 1970s. Pivotal for the issues that this article (...) addresses are Ian Nairn’s Outrage: On the Disfigurement of Town and Countryside, Robin Boyd’s The Australian Ugliness, Donald Gazzard’s Australian Outrage: The Decay of a Visual Environment, and the way the phenomenon of urban expansion is treated in these books in comparison with other books from the four national contexts under study, such as Ludovico Quaroni’s La torre di Babele and Reyner Banham’s The New Brutalism: Ethic or Aesthetic?. Particular emphasis is placed on Boyd’s articles in The Architectural Review between 1951 and 1970. At the core of the article is the analysis of the debates around ugliness between the 1950s and 1970s within the British, Italian, American and Australian contexts. (shrink)

Explications of the reconstruction of Leibniz’s metaphysics that Deleuze undertakes in 'The Fold: Leibniz and the Baroque' focus predominantly on the role of the infinitesimal calculus developed by Leibniz.1 While not underestimat- ing the importance of the infinitesimal calculus and the law of continuity as reflected in the calculus of infinite series to any understanding of Leibniz’s metaphysics and to Deleuze’s reconstruction of it in The Fold, what I propose to examine in this paper is the role played by other (...) developments in mathematics that Deleuze draws upon, including those made by a number of Leibniz’s near contemporaries – the projective geometry that has its roots in the work of Desargues (1591–1661) and the ‘proto-topology’ that appears in the work of Du ̈rer (1471–1528) – and a number of the subsequent developments in these fields of mathematics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. What is provided in this paper is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Role of the Heavens in the Thought of Philip MelanchthonCharlotte MethuenPhilip Melanchthon has long been recognized as one of the central figures in the German Lutheran Reformation. His theological contribution to the Reformation may be found in his codifying of Lutheran theology in the Confessio Augustana and in the Loci Communes, the first major Lutheran theological textbook, which long remained a central text for the teaching of theology (...) in Lutheran schools and universities. A teacher at the University of Wittenberg from 1518 until his death, Melanchthon was also involved in the reform of universities and the establishment of school systems throughout Lutheran Germany. These reforms facilitated the spread of the Reformation by encouraging the development of Lutheran educational principles and the establishing of Humanist education as the basis for the schools and universities of Lutheran Germany, and Melanchthon’s role in this process has led to his being accorded the title Praeceptor Germaniae. 1 More recently, Melanchthon has also been hailed as the initiator of a new “Lutheran” approach to natural philosophy. 2 It is my contention in this article that Melanchthon’s interest in natural philosophy is integrally connected to his educational interest and that these hang upon a particular theological and cosmological [End Page 385] understanding of the universe, particularly the heavens, as becomes clear from his defense of astronomy.I. Despite his theological interests and his commitment to the dissemination of Lutheran theology, Melanchthon was reluctant to teach theology and remained primarily concerned with the teaching of the subjects which made up the curriculum of the arts faculty in a sixteenth-century university. It was as Professor of Greek that he was originally appointed to the University of Wittenberg, and he had a deep interest in the teaching of ancient languages, especially Greek, and in the study of ancient texts in their original form rather than through later commentators. His promotion of the trivium (grammar, logic, and rhetoric) led him to write a textbook in each of these areas; and these works continued to be used as prescribed works in many German universities into the seventeenth century. 3 As an educator Melanchthon was thus firmly in the humanist mold. However, Melanchthon’s interests did not only take in the teaching of languages; he also promoted the teaching of both moral and natural philosophy. 4 Moreover, he was a strong advocate of the teaching of the mathematical sciences, particularly arithmetic, geometry, and astronomy. These, together with music, made up the quadrivium, which complemented the linguistic arts.In Melanchthon’s day the mathematical sciences had long been accepted as part of the arts curriculum, 5 even if in practice minimal attention had been paid them at many universities. 6 Their ambiguous status is reflected by the attitude taken by many humanist reformers, who did not regard arithmetic, geometry, astronomy, and music as an essential part of the curriculum. Thus Erasmus believed that music, arithmetic, and geography should be taught only to pupils who had a particular interest in these subjects 7 and that knowledge of nature, astrology, architecture, and other related subjects would come from reading the ancient authors. 8 In a similar vein Juan Luis [End Page 386] Vives argued that mathematics tended to divorce the mind from the practical concerns of life and that its study was not, therefore, to be recommended. 9 In contrast Melanchthon seems to have recognized the value of the study of the mathematical sciences from the beginning of his scholarly career. Perhaps he was inspired by his professor of mathematics at Tübingen, Johannes Stöffler, a gifted mathematician who wrote his own Arithmetic and was also fluent in the praise of his subject. 10 In an early oration, De artibus liberalis, delivered in Tübingen in 1517, Melanchthon praises arithmetic as a principal part of philosophy; geometry for its role in producing the wonderful machines of war and the beautiful ornaments of cities; music for its elegance and harmony; and astronomy (very briefly) for its applications in medicine. 11This oration should almost certainly be read as part of the tradition of orations and odes which extolled each of the liberal arts for its practical (and often also... (shrink)

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto (...) intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. (shrink)

In this article we move beyond the problematic distinction between ‘higher’ and ‘lower’ cognition by accounting for so-called ‘higher’ cognitive capacities in terms of skillful activities in practices, and in terms of the affordances exploited in those practices. Through ethnographic research we aim to further develop the new notion of skilled intentionality by turning to the phenomenon of the tendency towards an optimal grip on a situation in real-life situations in the field of architecture. Tending towards an optimal grip (...) is an inherently affective and dynamic phenomenon. It has been under explored in philosophy, despite its central place in Merleau-Ponty’s phenomenology. When the architects experience discontent, attention is typically solicited by the affordances through which possible ways of continuing the improvement of the design can be explored. In various ways this tension or disequilibrium - fueling a tendency towards an optimal grip - is encountered in the complex architectural design practices that we observed. Based on our field material we distinguish between: a) grip in visual perception, b) grip on the design, and c) grip on ‘how to design’. The move towards an optimal grip takes shape by joining forces with a landscape of affordances. This landscape includes, for instance, affordances for grasping a cup of coffee, affordances provided by a cardboard model to improve it, as well as for engaging with a person who is not physically present. Because they are skilled, the architects can be responsive to these different affordances in similar ways. Furthermore, in practice the architects are responsive to multiple affordances simultaneously. Through our analysis of the tendency towards an optimal grip we show that our surroundings contribute to skillful action and cognition in a far more complex way than is currently acknowledged in philosophy and cognitive science. (shrink)

'...deserves serious attention among new theories in architecture, and is recommended for all university architectural collections.'|s CHOICE.