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This handbook provides readers with a well-illustrated and readable comparative guide to proportion systems in architecture, setting out the mathematical principles that underlie the main systems and illustrating these with examples of their use in historical and modern buildings. The main body of the text traces the interplay of abstraction and empathy through the history of science, philosophy and architecture from the early Greeks through to the two early twentieth-century architects who made proportion the focus of their work: Le Corbusier and Van der Laan. The book ends with a reflection on the present and future role of proportion in architecture.

This handbook provides readers with a well-illustrated and readable comparative guide to proportion systems in architecture, setting out the mathematical principles that underlie the main systems and illustrating these with examples of their use in historical and modern buildings. The main body of the text traces the interplay of abstraction and empathy through the history of science, philosophy and architecture from the early Greeks through to the two early twentieth-century architects who made proportion the focus of their work: Le Corbusier and Van der Laan. The book ends with a reflection on the present and future role of proportion in architecture.

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CONTENTS

List of figures = ⅷ

Preface = xi

Chapter 1 : The harmony of the world made manifest in Form and Number = 1

 1.1. Not with a bang, but a whimper = 1

 1.2. Critique of Wittkower = 6

 1.3. Building and knowing = 10

 1.4. The rebuilding of conviction = 13

Chapter 2 : Abstraction and empathy = 18

 2.1. Wilhelm Worringer = 18

 2.2. Paradoxical nature of the two categories = 22

 2.3. Empathy and abstraction in architectural proportion = 26

 2.4. Le Corbusier, classical architect = 28

 2.5. Dim Hans van der Laan, modern primitive = 33

Chapter 3 : Unit and multiplier = 40

 3.1. Order and complexity = 40

 3.2. Measuring and counting = 42

 3.3. Between one and two = 43

 3.4. Proportion systems as a unified field : geometry or number? = 45

 3.5. The source in geometry = 49

 3.6. The source in number = 51

 3.7. Architectural proportion and progress in mathematics = 55

Chapter 4 : The house as model for the universe = 58

 4.1. Circle and square = 58

 4.2. The house built of numbers = 62

 4.3. Pythagorean mathematics and architectural proportion = 67

 4.4. The house on fire = 70

 4.5. Attempts to rebuild the house = 74

Chapter 5 : The proportions of the Parthenon = 80

 5.1. Numerical and geometrical interpretations = 80

 5.2. Hambidge and dynamic symmetry = 84

 5.3. Hambidge's Parthenon = 87

 5.4. The Piraeus arsenal = 91

 5.5. Interpretations of the Parthenon based on whole-number ratios = 93

 5.6. Further development of the whole-number interpretation = 94

Chapter 6 : Plato : order out of chaos = 99

 6.1. The original and the copy = 99

 6.2. God as arranger = 105

 6.3. The proportional mean = 106

 6.4. Making the World's Soul = 109

 6.5. Making the World's Body = 111

Chapter 7 : Aristotle : change, continuity, and the unit = 116

 7.1. 'Sight is the principal source of knowledge' = 116

 7.2. The real is the individual = 121

 7.3. Change and its causes = 123

 7.4. Continuity and infinity = 127

 7.5. The unit = 130

 7.6. Architectural proportion and Aristotle's concept of measure = 132

Chapter 8 : Euclid : the golden section and the five regular solids = 137

 8.1. The architectural ratios inherent in Euclidean space = 137

 8.2. The theorem of Pythagoras and the principal ratios used in architectural proportion = 140

 8.3. The golden section : Euclid's proofs from the square = 143

 8.4. The golden section : Euclid's proofs from the decagon and star-pentagon = 147

 8.5. The five regular solids = 149

Chapter 9 : Vitruvius = 156

 9.1. Disputed value of the Ten Books on Architecture = 156

 9.2. Van der Laan’s interpretation of the fundamental principles = 158

 9.3. The symmetry of temples and of the human body = 162

 9.4. Body, circle and square = 167

 9.5. Other proportions prescribed in the Ten Books, including the square root of two = 169

 9.6. The system of proportions as a law of growth = 171

Chapter 10 : Gothic proportions

 10.1. The continuity of Classical culture and the legacy of Plato and Vitruvius = 173

 10.2. The practical geometry of the master masons = 179

 10.3. The Milan cathedral controversy = 181

 10.4. The proportions of Chartres cathedral = 184

 10.5. Otto von Simson = 185

 10.6. John James = 189

 10.7. Towards a simpler and more comprehensive solution = 192

 10.8. The recovery of Aristotle by the West = 197

 10.9. Knowing and measuring = 199

 10.10. Gothic architecture and Scholasticism = 202

Chapter 11 : Humanism and architecture = 207

 11.1. The individual focus = 207

 11.2. Proportion in perspective = 214

 11.3. Alberti on the art of building = 219

 11.4. San Sebastiano, Mantua = 225

 11.5. The villas of Palladio = 232

Chapter 12 : Renaissance cosmology = 237

 12.1. Empathic and abstract tendencies = 237

 12.2. Learned ignorance = 238

 12.3. The Copernican revolution = 240

 12.4. The infinite universe and the infinity of worlds = 244

 12.5. Johannes Kepler = 246

 12.6. The regular polyhedra = 248

 12.7. The music of the spheres = 251

Chapter 13 : The world as a machine = 253

 13.1. Rudolf Wittkower and the collapse of universal values = 253

 13.2. The atomist universe = 256

 13.3. The changed nature of mathematical proportion = 258

 13.4. Science and art = 261

 13.5. Breaking the bond between science and art = 266

Chapter 14 : From the outer to the inner world = 273

 14.1. Clearing the ground of the obstacles to knowledge = 273

 14.2. The duplication of worlds = 276

 14.3. The dissolution of mind = 280

 14.4. Passion replaces reason = 283

 14.5. Berkeley and Hume on proportion = 285

 14.6. Edmund Burke's attack on the theory of proportion = 290

 14.7. The rational imperative = 293

 14.8. Kant : the understanding as the architect of nature = 298

Chapter 15 : The golden section and the golden module = 304

 15.1. The rediscovery of the golden section = 304

 15.2. Adolf Zeising = 305

 15.3. Gustav Fechner = 307

 15.4. The statistical approach to the golden section = 308

 15.5. Attempts to explain the phenomenon = 313

 15.6. The curves of life = 313

 15.7. The golden section in Le Corbusier's early work = 316

 15.8. Origins and aims of the modulor = 321

 15.9. The geometry of the modulor = 323

 15.10. The modulor's arithmetic = 325

 15.11. The anthropometrics of the modular = 329

 15.12. The modular in practice : the Unit$$e'$$ d'habitation = 332

Chapter 16 : The house as a frame for living and a discipline for thought = 336

 16.1. Architecture : a practical or a speculative art? = 336

 16.2. Physical versus intellectual functions = 338

 16.3. The abstract revolution = 343

 16.4. The necessary and the essential = 345

 16.5. The search for a starting point = 348

 16.6. The intrinsic meaning of architecture = 351

 16.7. The first building = 354

 16.8. Articulate building = 356

 16.9. What system of proportion do we need? = 359

 16.10. Knowing by not knowing = 362

 16.11. Types and orders of size = 364

 16.12. Conclusion = 369

References = 373

Index = 383