- Proportion: Richard Padovan
- Chapter 1: Harmony of the World
- 1.1 Not with a bang, but with a whimper
- K.R. Popper: Cosmology is the problem of understanding the word-including ourselves, and our knowledge, as part of the world
- A. Einstein comments on Modulor: "makes the bad difficult and the good easy"
- By 1957 proportion had become an aesthetic matter
- E. Maxwell Fry
- W.E.Tatton Brown

- E. Maxwell Fry
- Making shapes that were pleasing
- The aesthetic argument for proportion becomes quickly entangled in contradictions.

- The aesthetic argument for proportion becomes quickly entangled in contradictions.
- Ignored the cosmological goodness
- View of architectural proportion held up to and during the Renaissance
- Beauty is a sign of something profound
- Accord with general harmony of the world
- Alberti called such order concinnitas
- Nature's chief concern is that whatever she produces should be absolutely perfect

- Nature's chief concern is that whatever she produces should be absolutely perfect
- Wittkower: Man-created harmony was a visible echo of a celestial and universally valid harmony

- Beauty is a sign of something profound

- K.R. Popper: Cosmology is the problem of understanding the word-including ourselves, and our knowledge, as part of the world
- 1.2 Critique of Wittkower
- Wittkower: We cannot find a position of belief (in proportion) as individuals because a broader foundation is lacking
- Proportion as it is currently regarded is atypical; individual subjective judgment
- The disregarding of proportion, for Wittkower, is due to the collapse of the cosmological foundation as a result of the Galilean and newtownian revolution
- Padovan differs in 3 respects
- 1. The decline of numerical systems of proportion began in architecture more than a century before Newton published his laws
- Newtonian science coincided with revival of proportion systems
- No conception of nature has been more perfectly harmonious and unified than Newton's equation for gravity
- Wittkower implies, but does not make the argument that the increasingly secular functions of buildings destroyed the mathematical basis of proportion

- Newtonian science coincided with revival of proportion systems
- 2. Documented examples of the direct impact on architecture of discoveries in mathematics or science are scarce
- Pythagorean harmonies influencing the Italian Renaissance is the exception, not the rule
- Attempts to prove pythagorean mathematics influenced Greek architecture are inconclusive
**The mathematics of architectural proportion has always been of a kind suited to the forms and techniques of building, ane these have changed little in their essentials, even today**

- Pythagorean harmonies influencing the Italian Renaissance is the exception, not the rule
- 3. What finally placed a gulf between the human mind and the external world were the doubts raised by Hume and empircist philosophy
- Not a development is science
- Mathematics were discoverable by pure thought but had no necessary relation to the external world
- Undermining of all certainty in human knowledge

- Not a development is science

**Leitmotiv: Without waiting, passively, for repetitions to impress or impose regularities upon us, we actively try to impose regularities on the world.**

- Wittkower: We cannot find a position of belief (in proportion) as individuals because a broader foundation is lacking
- 1.3 Building and knowing
- Systems of proportion enable our buildings to embody a mathematical order the we either distil out of or impose upon nature
- Secondary concerns
- Modular coordination allows building components to fit together without waste
- Pleasing visual harmony

- Modular coordination allows building components to fit together without waste

**Parallels between measure in architecture and the mathematics of nature is the basic aim of this study**

- Two opposing views of mathematics
- 1. The mathematical order that we discover more or less imperfectly manifested in phenomena constitutes their true or essential nature
- 'Empathy'

- 'Empathy'
- 2. Mathematics is a purely artificial construction, a system of conventional signs and the rule for manipulating them
- All assertions about the ultimate nature of the universe, mathematical or not, are rejected as 'metaphysical'
- 'Abstraction'

- All assertions about the ultimate nature of the universe, mathematical or not, are rejected as 'metaphysical'

- 1.4 Rebuilding of conviction
- Van der Laan
- Proportion is a regularity imposed by the mind upon the world
- Deeply rooted in Greek Philosophy
- False understanding of Greek philosophy by Padovan

- False understanding of Greek philosophy by Padovan
- Intrinsic seperateness of man and nature

- Proportion is a regularity imposed by the mind upon the world
- Le Corbusier
- Ignores the disconnection asserted to have developed in 1700 between science and art
- "...man is no longer the operative force, but rather it is his contact with the universe"
- Intrinsic unity of man's thinking and building with the natural world

- Ignores the disconnection asserted to have developed in 1700 between science and art

- Chapter 2: Abstraction and Empathy
- 2.1 Wilhelm Worringer
- Proportion systems reflect a 'weltanschauung'
- 3. Oriental
- Architectural proportion has been affected by the two schools of thought about human knowledge
- Empathy
- To know is to belong to it
- We know what we are of
- 'in-feeling'
- Unity
- Can emerge only where men feel completely at home in the external world
- 1. Classical

- To know is to belong to it
- Abstraction
- 2. Primitive
- To know is to have made it
- Mathematics is our own creation
- To draw away from
- Duality
- Protection or refuge

- 2. Primitive

- Abstraction and Empathy: A contribution to the Psychology of style
- Two clear categories of art, corresponding to two opposed human attitudes to the external world

- Two clear categories of art, corresponding to two opposed human attitudes to the external world

- Proportion systems reflect a 'weltanschauung'
- 2.2 Paradoxical nature of the two categories
- Empathy
- Humanization of nature rather than the imitation

- Humanization of nature rather than the imitation
- Abstraction
- Meditation on and a distillation from nature

- Meditation on and a distillation from nature
- Inside every empathist there is an abstractionist fighting to get out,, and vice versa

- 2.3 Empathy and abstraction on architectural proportion
- The same division Worringer makes between empathy and abstraction in representational art applies equally to the relation of of architectural proportion to nature's laws
- Empathy
- Serene reflection or elucidation of the underlying harmony of nature

- Serene reflection or elucidation of the underlying harmony of nature
- Abstraction
- Protest or reaction against nature's overwhelming formlessness, caprice, obscurity

- Protest or reaction against nature's overwhelming formlessness, caprice, obscurity

- The same division Worringer makes between empathy and abstraction in representational art applies equally to the relation of of architectural proportion to nature's laws
- 2.4 Le Corbusier, Classical architect
- Empathy
- Representative of Classical tradition of looking to the cosmos for the ultimate justification of mathematical proportions in art
- Laws of nature are the laws of 'our nature'
- Behrens
- Nature is recast to fit mans mathematical scheme
- Nature appears as chaos; but nature is animated by a spirit of order
- The conflict is resolved much as Plato resolved it: by distinguishing between appearances and knowledge

- The conflict is resolved much as Plato resolved it: by distinguishing between appearances and knowledge

- Empathy
- 2.5 Dom Hans Van Der Laan; Modern Primitive
- Starting out from man's need to construct his own artificial, delimited world within the limitless world of nature, comes to the conclusion that this little man-made world is a necessary conclusion of the natural process as such
- Empathy and abstraction each contain an intimation of its opposite

- Starting out from man's need to construct his own artificial, delimited world within the limitless world of nature, comes to the conclusion that this little man-made world is a necessary conclusion of the natural process as such

- Chapter3: Unit and Multiplier
- 3.1 Order and complexity
- Absence of order is confusion
- Absence of complexity is disorder
- Proportion systems can be described as the creation of an ordered complexity

- Absence of order is confusion
- 3.2 Measuring and counting
- The history of proportion in architecture could be told in terms of solutions of the problem of resolving the conflict between these two operations
- Pythagoreans taught that the world was composed of individual units
- Space was countable, and therefore measurable
- The incommensurability of the side of a square with its diagonal shattered the Pthagoreans conception

- Space was countable, and therefore measurable
- Aristotle holds the measure of number as the most accurate
- The individual unit is something chosen, but must not be too big or too little

- The history of proportion in architecture could be told in terms of solutions of the problem of resolving the conflict between these two operations
- 3.3 Between one and two
- The beginning of proportion is the passage from one two
- Adding one to itself; three ways of proceeding
- 1. Process of addition: 1,2,3,4,5 arithemtic progression
- 2. Multiply each number by a constant: 1,2,4,8,16 geometric progression, embryonic system of proportion
- In practice, the builder or architect needs to be able to employ ratios that are greater than 1:1, but less than 2:1
- For instance, he must be able to make a room larger than a square but shorter than two squares

- For instance, he must be able to make a room larger than a square but shorter than two squares

- 3. Interweave two progressions

- 1. Process of addition: 1,2,3,4,5 arithemtic progression

- The beginning of proportion is the passage from one two
- 3.4 Proportion systems as a unified field: geometry or number?
- Most proportion systems can be generated by simple geometry
- All systems can be generated as additive series of whole numbers
- The pattern can manifest itself either as goemetry or as number

- Most proportion systems can be generated by simple geometry
- 3.5 The source in geometry
- Square=root 2
- Equilateral triangle, half base to height= root 3
- Regular pentagon, diagonal to side= golden section
- Regular hexagon, height to side= root 3
- Regular decagon, radius to side= golden section
- Double square= root 4
- Double square inscribed in semi circle= root 2
- Square inscribed in semicircle, diameter to side of square= root 5
- 5 regular polyhedra
- Regular tetrahedron
- Cube
- Regular octahedron
- Regualar icosahedron
- Regular dodecahedron

- Regular tetrahedron

- Square=root 2
- 3.6 The source in number
- The more additive permutations a system offers, the greater the order
- It is no less valid to regard proportion systems as additive progressions of whole numbers that happen to converge towards irrational values, than as progressions of irrational numbers that arise from geometrical constructions such as the diagonal of the square

- The more additive permutations a system offers, the greater the order
- 3.7 Architectural proportion and progress in mathematics
- All known systems of proportion are solutions to the problem of how to punctuate the interval between 1 and 2 in such a way that the resulting series of measures is both additive and multiplicative
- Developments in the field of architectural proportion has not been comparable to the progress of the physical sciences
- Architectural proportion has always needed to suit the forms and techniques of building
- Mathematics of architectural proportion has remained more or less that of the ancient Greeks

- All known systems of proportion are solutions to the problem of how to punctuate the interval between 1 and 2 in such a way that the resulting series of measures is both additive and multiplicative

- Chapter 4: The house as a model for the universe
- 4.1 Circle and square
- To know something, one must have made it oneself
- Man used the same geometry in building and cosmology: the circle and the square
- Almost all the irrational numbers numbers important for architectural proportion are generated by simple constructions involving the circle, square, and right angled triangle

- To know something, one must have made it oneself
- 4.2 The house built of numbers
- 1,2,3,4 = 10
- Pythagoreans appeared to have unified arithmetic and geometry
- Ontological and concerned with being

- Ontological and concerned with being
- The unit or point as the smallest indivisible whole
- The irrational number threatened Pythagorean mathematics

- 1,2,3,4 = 10
- 4.3 Pythagorean mathematics and architectural proportion
- Pythagorean-Platonic
- Geometry
- Root 2, Root 3, Root 5
- Theory that medieval buildings were set out geometrically using arcs and circles, is unproven

- Geometry
- Renaissance-Classical
- Arithmetical
- 2:1, 3:2, 4:3

- Arithmetical

- 4.4 The house on fire
- Architectural metaphor of the house as image of the world
- If architectural proportion has a basis in nature, the first step towards its discovery is to realize 'nature loves to hide'

- Architectural metaphor of the house as image of the world
- 4.5 Attempts to rebuild the house
- Paradox of continuity and discreetness
- Democritus' theory of knowledge make him seem an extraordinarily accurate forerunner of the scientific revolution 1600-1900

- Paradox of continuity and discreetness

- Chapter 9: Vitruvius
- 9.1 Disputed value of the Ten Books of architecture
- The obscurity of Vitruvius' terminology has presented a problem for all his interpreters
- Van der Laan gives a different but remarkably consistent interpretation, while Wittkower concludes desparingly that 'Vitruvius' work contains no real theory of proportion'

- The obscurity of Vitruvius' terminology has presented a problem for all his interpreters
- 9.2 Van der Laan's interpretation of the fundamental principles
- Vitruvius' six fundamental principles of architecture
- 1. Ordinatio, taxis, or ordannance
- 2. Dispositio, diathesis, or disposition
- 3. Eurythmia
- 4. Symmetria
- 5. Decor or decorum
- 6. Distributio, oikonomia or economy

- 1. Ordinatio, taxis, or ordannance
- Van der Laan views two main principles and four subdivisions
- Ordonnance
- Symmetry
- Denotes the ratios between corresponding measures of different things
- Scale

- Denotes the ratios between corresponding measures of different things
- Eurhythmy
- Denotes the ratios between the different measures of the same thing
- Shape

- Denotes the ratios between the different measures of the same thing

- Disposition
- Decorum
- Conventional association of style with meaning

- Conventional association of style with meaning
- Economy
- Practical arrangement

- Practical arrangement

- Van der Laan says the fundamental aim of art is to connect the two poles of ordonnance and disposition, or material and intellect

- 9.3 The symmetry of temples and of the human body
- Symmetry / Proportion has three requirements
- 1. The measures of all the parts and of the whole must agree or correspond with each other
- 2. There must be a direct relation between the whole or largest measure and an elementary part or module
- 3. It follows from 1 and 2 that the measures of all or other parts must likewise relate both to the whole and the module

- 1. The measures of all the parts and of the whole must agree or correspond with each other
- Here we have one of the least deniable, because most practical, explanations of the need for proportion in architecture: it is demanded by the basic necessity of putting buildings together out of a number of seperate bits, and of these bits fitting each other

- 9.4 Body, circle, square
- Vitruvian man has been influential
- Mathematically most successful descendant of vitruvian man is Le Corbusier's modular man

- Vitruvian man has been influential
- 9.5 Other proportions prescribed in the Ten Books, including the square root of two
- Vitruvius describes war machines in terms of mathematical proportions
- The lengths of temples should always be twice their widths
- The inclusion of an incommensurable ratio, root 2, is an apparent anomaly among the exclusively rational proprtions used elsewhere

- Vitruvius describes war machines in terms of mathematical proportions
- 9.6 The system of proportions as a law of growth
- Contrary to modern assumptions, the calculation of mathematical proportions comes at the very start of the design process, not at the end as a final polish

- Contrary to modern assumptions, the calculation of mathematical proportions comes at the very start of the design process, not at the end as a final polish

- Chapter 5: The proportions of the parthenon
- 5.1 Numerical and geometrical interpretations
- Viollet-le-duc
- Favors two triangles in geometric analysis
- 'Egyptian'
- Equilateral

- 'Egyptian'

- F.M. Lund
- Three sets of pentagons

- Three sets of pentagons
- Viollet-le-duc and F.M.Lund imposed arbitrary geometric constructions

- 5.2 Hambidge and dynamic symmetry
- Static symmetry
- Dynamic symmetry

- Static symmetry
- 5.3 Hambidge's Parthenon
- Reduces the Parthenon to three geometrically interrelated figures

- Reduces the Parthenon to three geometrically interrelated figures
- 5.4 The piraeus arsenal
- Hambidge's interpretation is unsupported by textual evidence about the methods followed by Greek architects
- Two remarkable things
- Based on simple commensurable rations like 1:2, 2:3
- The ideal ratios are rounded up or down where necessary in order to retain whole number measures in feet

- Based on simple commensurable rations like 1:2, 2:3

- Hambidge's interpretation is unsupported by textual evidence about the methods followed by Greek architects
- 5.5 Interpretations of the Parthenon based on whole-number ratios
- Much simpler than Hambidge is A.W.Lawrence
- The test of a system of proportions is thus whether it generates a sequence of different but related raios
- A.W. Lawrence points out the 9:4 ratio in the most important design elements
- 1. Ratio of plane to stylobate
- 2. Ratio of the temple front-width of stylobate to column plus entablature
- 3. Ratio of column interval to the lower diameter of column
- A.W. Lawrence does not account for 2 things
- 1. Ratio of total height to the column interval-determines structural bay
- 2. Ratio of the column height to entablature

- 1. Ratio of total height to the column interval-determines structural bay

- 1. Ratio of plane to stylobate

- Much simpler than Hambidge is A.W.Lawrence
- 5.6 Further development of the whole-number interpretation
- What relation can be found between the column interval and the larger set of dimensions?
- An analysis based on 42 digits has the advantage of producing an unbroken chain of commensurable ratios
- No final answer to the Parthenon's proportional problem

- What relation can be found between the column interval and the larger set of dimensions?

- Chapter 6: Plato - order out of chaos
- 6.1 The original and the copy
- Intellectual and sensible aspects of experience
- Aristotle and Plato synthesised intellectual and sensible
- Plato
- Geometrization of arithmetic, astronomy, and cosmology
- Basis of Plato's synthesis is the notion of the image
- The relation of the Forms to their sensible images is likened to a proportion
- Sees the mathematical realities not as a human projection onto nature, but as an essential property of nature itself
- The human soul has access to knowledge because it originates from the same source

- Geometrization of arithmetic, astronomy, and cosmology
- Aristotle

- Intellectual and sensible aspects of experience
- 6.2 God as arranger
- Plato presents his cosmology in the form of a cosmogony
- The demiurge rearranges the world to bring it from disorder into order
- The world is a living being like ourselves

- Plato presents his cosmology in the form of a cosmogony
- 6.3 The proportional mean
- Two things, called the extremes, are united by a third, the mean
- The function of proportion in binding things together
- Proportion binds together the parts of Plato's world just as it binds together the parts of a building.
- It is a matter of construction: of uniting seperate elements to make an integrated whole
- 1. Geometric
- 2. Arithmetic
- 3. Harmonic
- 4. Contraharmonic

- Two things, called the extremes, are united by a third, the mean
- 6.4 Making the world's soul
- Pythagorean mathematics
- 1+2+3+4+9+8=27
- 1 4/3 3/2 3/2 2/1 2 3 8/3 9/2 3/1 6/1 4 9 16/3 27/2 6/1 18/1 8 27
- Produces mutually commenurable dimensions, and shapes in plan and elevation comppsed of arrangements of squares

- Pythagorean mathematics
- 6.5 Making the world's body
- Square roots that are not square numbers (1, 4, 9, 16) will be irrational or incommensurable as lengths, commensurable as areas.
- Two contrasting aspects of early Greek mathematics and two possible approaches to the problem of architectural proportion
- 1. Whole number ratios derived from the Pythagorean theories of musical harmony and of numbers as finite monads arranged in various patterns to constitute the visible world
- 2. The analysis of the first four regualre polyhedra as the primary atomic constituents of fire, earth, air, and water, Plato attmepts to resolve the paradox presented by the Pythagorean discovery of irrational numbers, and in so doing opens the way to the geometrization of mathematics

- 1. Whole number ratios derived from the Pythagorean theories of musical harmony and of numbers as finite monads arranged in various patterns to constitute the visible world

- Square roots that are not square numbers (1, 4, 9, 16) will be irrational or incommensurable as lengths, commensurable as areas.

- Chpater 7: Aristotle - change, continuity, and unit
- 7.1 'Sight is the principal source of knowledge'
- In contrast to the rigorously quantitative method that has characterised Western science since the seventeeth century, he approches things in terms of their essential natures
- Aristotle's philisophical empiricism, preciesly because of its contridictions, would lead in the later middle ages to a critical process that became the basis of the experimental tradtition of modern science

- In contrast to the rigorously quantitative method that has characterised Western science since the seventeeth century, he approches things in terms of their essential natures
- 7.2 The real is the individual
- Aristotle does not deny the reality of universals altogether
- Species and Genera remain 'substances', even if secondary and dependant ones
- His fence sitting on this point sowed the seeds of the disputes between nominalists and realists that rocked Medieval Scholasticism

- Aristotle does not deny the reality of universals altogether
- 7.3 Change and it's causes
- The individual being is subject to change: it moves, it is born and grows, it decays and dies.
- Universe is composed of two seperate regions
- 1. Sublunar sphere and the four elements
- 2. Celestial sphere with constant celestial motion that is therefore changeless

- 1. Sublunar sphere and the four elements
- Four kinds of cause
- 1. The material out of which a thing comes into being, such as the bronze of a statue (traditionally called the material cause)
- 2. The form or pattern of a thing (formal cause)
- 3. The person or thing that sets the change in motion - for example, the maker is the cause of the thing made (the efficient cause)
- 4. The end or goal - that is, the purpose for which a thing exists or is made (the final cause)

- 1. The material out of which a thing comes into being, such as the bronze of a statue (traditionally called the material cause)
- Two principles basic to his conception of change
- 1. Actuality
- 2. Potentiality

- 1. Actuality
- The end is implicit in the beginning
- Matter is not simply like the steel of which the spring is made; it is like the coiled spring in which the latent power of movement is stored
- In this way the Platonic Form of the species is brought down from its heaven of unchanging reality, and plunged in the flow of time and sensible existence
- A living organism remains 'itself' even though its constituent matter is constantly beign renewed by the absorption of nutrients and the elimination of waste products

- The individual being is subject to change: it moves, it is born and grows, it decays and dies.
- 7.4 Continuity and infinity
- Two kinds of infinity
- 1. By addition
- 2. By division

- 1. By addition
- Only procecesses can be infinte, not actualities
- Continuity versus discreete parts

- 7.5 The unit
- Various forms that unity can take, to be 'one'
- 1. To be essentially a 'this' and seperate either in place or form or in thought
- 2. To be a whole and indivisible
- 3. Above all it is to be the first measure of a class - especially of quantity, from which it has been extended to other catagories

- 1. To be essentially a 'this' and seperate either in place or form or in thought
- The unit of measuer must be appropriate not only to the quantity of the thing measured - it must be neither too big nor too small - but also to its quality

- 7.6 Architectural proportion and Aristotle's concept of measure
- Fundamental unity of all systems of proportion, and their common grounding in reflections upon the nature of the world and our human relation to it
- Consider a series of volumes whose unit of measure is the cube
- Proportional systems can generate volumes

- Fundamental unity of all systems of proportion, and their common grounding in reflections upon the nature of the world and our human relation to it

- Chapter 8: Euclid - the golden section and 5 regular solids
- 8.1 The architectural ratios inherent in Euclidian space
- Golden section
- Geometry of the 5 regular solids

- Golden section
- 8.2 The theorem of Pythagoras and the principal ratios used in architectural proportion
- Root 2
- Root 3
- Root 5
- Golden Sedtion

- Root 2
- 8.3 The golden section: Euclid's proofs from the square
- The proof demonstrates that as the given line is to the larger segment, so is the larger segment to the smaller

- The proof demonstrates that as the given line is to the larger segment, so is the larger segment to the smaller
- 8.4 The golden section: Euclid's proofs from the decagon and star-pentagon
- The proposition suggests two simple constructions which relate together the sides of all the principal polygons

- The proposition suggests two simple constructions which relate together the sides of all the principal polygons
- 8.5 The five regular solids
- All five regular polyhedra are united geometrically by the fact that they can be inscribed more or less simply within each other
- No more than 5 regular polyhedra can exist
- Euclidian geometry has been replaced by the possibility of inventing seperate self-consistent geometries appropriate to particular purposes and different scales
- All geometries are not something found in nature, once and for all, but man made frames of reference, valid for certain purposes and unsuited for others

- All five regular polyhedra are united geometrically by the fact that they can be inscribed more or less simply within each other

- Chapter 10: Gothic Proportions
- 10.1 The continuity of Classical culture and the legacy of Plato and Vitruvius
- Full contact did not begin to be restored for at least seven hundred years - first indirectly, through the mediation of Arabic translations after about 1100, and eventually by the influx of Greek scholars and the acquisition of original Greek texts that followed the fall of Constantinople in 1453
- Thus is was increasingly the Pythagorean - that is, the numerical and mystical - aspects of Plato's dialogue that drew most attention
- For Augustine, Plato's account of the creation in the Timaeus was a wonderful anticipation of Christian teaching

- Full contact did not begin to be restored for at least seven hundred years - first indirectly, through the mediation of Arabic translations after about 1100, and eventually by the influx of Greek scholars and the acquisition of original Greek texts that followed the fall of Constantinople in 1453
- 10.2 The practical geometry of the master masons
- German Master Builder, Mathes Roriczer
- 1. Booklet on Pinnacles
- 2. German Geometry
- 1. Method for constructing Pentagon
- 2. Method for constructing Heptagon
- 3. Method for constructing Octagon
- 4. Method for calculating the circumference of a circle of given diameter
- 5. Method for drawing a square and equilateral triangle of equal area

- 1. Method for constructing Pentagon
- First stages of Roriczer's method for designign pinnacles
- 1. Draw a square
- 2. Inscribe it within a smaller square, the corners of which are the midpoints of the first
- 3. Repeat the operation
- 4. Rotate the second square 45 Deg so that all three are parallel and concentric, and have a constant root 2 ratio between their widths
- 5. The elevations are then derived by extrapolation from the plan

- 1. Draw a square
- Unaware of Euclid's Elements and simple and exact constructions of the pentagon

- 1. Booklet on Pinnacles

- 10.3 The Milan cathedral constroversy
- The medieval master mason seems to have regarded proportion as a sure guide to structural stability
- If the proportion was sound, so would be the structure
- In Plato's words, proportion is literally the bond that holds things together
- Nicolas de Bonaventure proposed a proportion of 8:5
- All dimensions proposed were perfectly commensurable
- All dimensions are suited to measurement rather than geometrical setting out

- All dimensions proposed were perfectly commensurable
- Annas de Firimburg proposed a proportion of root 3:1
- Gabriele Stornaloco sustituted a rational 7:4 to replace the root 3:1
- Heinrich Parler of Gmund proposed an ad quadratum scheme
- Undermines the assumption that Gothic proportion was the result of geometrical rather than metrical setting-out methods
- No one principle of proportion can be attributed with any confidence to any particular period

- The medieval master mason seems to have regarded proportion as a sure guide to structural stability
- 10.4 The proportions of Chartres cathedral
- Otto von Simson in The Gothic Cathedral
- John James in The Contractors of Chartres

- Otto von Simson in The Gothic Cathedral
- 10.5 Otto von Simson
- The Gothic cathedral was designed as a model or image of the mathematical structure of the universe
- Finds the cathedral's proportions dominated by the golden section
- Odd conclusion given the two major influences on the design of Chartres Cathedral
- Augustinian - therefore musical and numerical

- Augustinian - therefore musical and numerical
- Two proportional givens
- 1. Cross section is an exact square
- 2. The longitudinal dimensions are all less than the lateral ones

- 1. Cross section is an exact square
- Medieval architect determined all dimensions geometrically on site using cords and pegs

- The Gothic cathedral was designed as a model or image of the mathematical structure of the universe
- 10.6 John James
- 1.The Cathedral was built in a succession of short campaigns by a series of different masters and their teams crafstmen
- 2. An almost sacred importance was attached by the Medieval architects to standard measures and to the use of whole units
- The irregularity of Gothic construction encountered in medieval buildings makes it virtually impossible to determine the proportional scheme from measurements, and that only the simplest relations. apparent to the eye, give any basis for judgment

- 1.The Cathedral was built in a succession of short campaigns by a series of different masters and their teams crafstmen
- 10.7 Towards a simpler and more comprehensive solution
- 1. The spirit in which the structure appears to be designed
- 2. The mathematical knowledge available to the builders
- 3. Technical feasibility and rationality of the setting-out operation demanded

- 1. The spirit in which the structure appears to be designed
- 10.8 The recovery of Aristotle by the West
- The intense period of Gothic cathedral building in northern France was concentrated in the sixty years 1170 to 1230
- St. Thomas Aquinas (1224-74) was not born until the end of this period, and his most important writings belong to 1263-73
- It is by way of Islamic Spain that Western Christendom was able to recover its lost contact with the sources of Greek Scholarship
- Plato's doctrine of immaterial forms sat comfortably with Christian doctrine - Aristotle's principle that reality lies in individual things and sight did not
- Aquinas was canonized in 1323, and by implication so was Aristotle

- The intense period of Gothic cathedral building in northern France was concentrated in the sixty years 1170 to 1230
- 10.9 Knowing and measuring
- Knowledge is a process of making or abstraction, not of belonging or empathy
- For Aquinas, the mind receives ideas by its own activity
- Aquinas's definition of truth as concordance between thing and intellect is remarkably close to later empiricism
- Aquinas sees the mind's idea of a thing as a sort of proportional mean between thing and intellect

- Knowledge is a process of making or abstraction, not of belonging or empathy
- 10.10 Gothic architecture and scholasticism
- Panofsky argues of a hardly accidental occurance between Gothic architecture and Scholasticism in the purely factual domain of time and place
- The Scholastic summa is composed in a way that conforms to Vitruvius' defintion of proportion in architecture
- As the same time as total unity was thus achieved by standardization of forms (rib vault instead of variety of vaults), the number of seperate parts was multiplied
- According to Panofsky the individual elements, while forming an indiscerptible whole, yet must proclaim their identity by remaining clearly seperated from each other
- Panofsky's three Gothic Problems
- 1. The rose window in the west facade
- 2. The organization of the wall below the clerestory
- 3. Conformation of the nave piers

- 1. The rose window in the west facade
- Panofsky's case rests on 3 premises
- 1. However self-evident a parallel may seem, we cannot be content 'if we cannot imagine how it came about'
- 2. The parallel between Gothic architecture and Scholastic philosophy 'is a genuine cause and effect relation', and not a mere prallelism
- 3. It is backed up by a concurrence in the purely factual domain of time and place

- 1. However self-evident a parallel may seem, we cannot be content 'if we cannot imagine how it came about'
- Architecture often antedates scholasticism by nearly eight decades
- Architecture influenced philosophy, not the reverse - architecture led the way for science

- Panofsky argues of a hardly accidental occurance between Gothic architecture and Scholasticism in the purely factual domain of time and place

- Chapter 11: Humanism and architecture
- 11.1 The individual focus
- Kristeller identifies 4 strands of Humanism
- 1. Humanism Proper (Francesco Petrarca and Lorenzo Valla)
- 2. Neoplatonism (Marsilio Ficino and Pico della Mirandola)
- 3. Aristotelianism (Poetro Pomponazzi)
- 4. Naturalism (Francesco Patrizi and Giordano Bruno)

- 1. Humanism Proper (Francesco Petrarca and Lorenzo Valla)
- Renaissance Humanism is indeed characterized by empathy, by like all empathy it carries within it an unconscious undercurrent of abstraction

- 11.2 Proportion in perspective
- Calculated proportions cannot be perceived as one moves around a building because they are distorted by perspective
- The Renaissance answer: the laws of perspective are ruled by proportion
- Painter's persepctive was invented by an architect
- The apparent sizes of a series of equal objects spaced at regular intervals from the eye form a harmonic progression

- Calculated proportions cannot be perceived as one moves around a building because they are distorted by perspective
- 11.3 Albert on the art of building
- Alberti is prescribing how the buildings of the future are to be built
- Contrary to Wittkower, Alberti uses irrational numbers and envisaged root 2 proportion being used

- Alberti is prescribing how the buildings of the future are to be built
- 11.4 San Sebastiano, Mantua
- If the proposed system of proportion is indded the basis of Alberti's design of S. Sebastiano, it indicates that the assumed chronology of the evolution of architectural proportion is in need of revision

- If the proposed system of proportion is indded the basis of Alberti's design of S. Sebastiano, it indicates that the assumed chronology of the evolution of architectural proportion is in need of revision
- 11.5 The villas of Palladio
- Those proportional relationships which other architects had harnessed for the two dimensions of a facade, or the three dimensions of a single room were employed by him to integrate a whole structure
- Palladio and the Humanist scholars were interested not only in whole number harmonic intervals, but also the ratios resulting from the irrational square roots of two and three and the golden section

- Those proportional relationships which other architects had harnessed for the two dimensions of a facade, or the three dimensions of a single room were employed by him to integrate a whole structure

- Chapter 12: Renaissance cosmology
- 12.1 Empathic and abstract tendencies
- Empathic
- Copernicus
- Kepler

- Copernicus
- Abstract
- Cusanus
- Bruno

- Cusanus

- 12.2 Learned ignorance
- Cusanus
- Denies the possibility of the mathematical treatment of nature
- Negative theology
- Infinity

- Denies the possibility of the mathematical treatment of nature

- 12.3 The Copernican revolution
- What drove him to seek alternative explanations was not scientific need but a conviction - the opposite of Cusanus' denial of the possibility of mathematical exactitude - that the universe must be a mathematically ordered harmony

- What drove him to seek alternative explanations was not scientific need but a conviction - the opposite of Cusanus' denial of the possibility of mathematical exactitude - that the universe must be a mathematically ordered harmony
- 12.4 The infinite universe and the infinity of worlds
- Bruno asserted that the universe had neither limits nor center

- Bruno asserted that the universe had neither limits nor center
- 12.5 Johannes Kepler
- Three mathematical laws
- 1. The planetary orbits are elliptical, not circular
- 2. The uneven velocity of the planets as they move around the sun is determined by the law that the line joining the centers of sun and planet sweeps out equal areas of the elliptical plane in equal time
- 3. The squares of the periodic times (that is, times to complete a revolution) of the planets are proportional to the cubes of their respective mean distances from the sun

- 1. The planetary orbits are elliptical, not circular
- His work can only be understood in the light of a world-view that is as much religious and aesthetic as it is purely physical

- 12.6 The regular polyhedra
- The regular polyhedra are used by Kepler in a cosmological model
- The ratios that determine the proportions of Kepler's universe are the same that governed Plato's world and the geometry of Euclid, and have traditionally been the basis of proportion in architecture
- Kepler's three laws were the happy result of mistaken preconceptions

- The regular polyhedra are used by Kepler in a cosmological model
- 12.7 The music of the spheres
- Kepler attempts to determine the relations of the planetary orbits on the basis of Platonic mathematic, but this time combining the geometry of regular polygons with the arithmetical intervals of the musical scale
- Physical and metaphysical were two synonymous words for Kepler

- Kepler attempts to determine the relations of the planetary orbits on the basis of Platonic mathematic, but this time combining the geometry of regular polygons with the arithmetical intervals of the musical scale

- Chapter 13: The world as machine
- 13.1 Rudolf Wittkower and the collapse of universal values
- Wittkower & Koyre:This implies the discarding by scientific thought of all consideration based upon value-concepts, such as perfection, harmony, meaning and aim, and finally the utter devalorization of being, the divorce of the worl of value from the world of facts
- The decline of Renaisssance canons of proportion in Mannerist architecture preceded any possible influence from Baroque science
- Although the new cosmology invlolved an infinite universe obeying 'mechanical laws', the scientists themselves did not regard their universe as being without an 'ulterior plan'
- Deism

- Wittkower & Koyre:This implies the discarding by scientific thought of all consideration based upon value-concepts, such as perfection, harmony, meaning and aim, and finally the utter devalorization of being, the divorce of the worl of value from the world of facts
- 13.2 The atomist universe
- Pierre Gassendi (1592-1655)
- The ancient theory of atomism anticipated the corpuscular-kinetic world-picture of Galileo and Newton
- Leucippus
- Democritus
- Epicurus

- Leucippus
- Milic Capek writes: The only difference between Greek atomism and nineteenth century physics was that the latter had incomparably more efficient and conceptual tools at its disposal than Democritus and Leucippus...Fundamentally the basic conceptions were the same. This was the deep historical reason why the birth of modern science occured simultaneously with the revival of atomism by Bruno, Bacon, Gassendi, and others
- The atomist universe composed of four distinct entities
- 1. Absolute space
- Homogeneous
- Continuous
- Infinitely extended
- Infinitely divisible
- Causally inert - it does not imply, nor can it affect or be affected by the matter contained in it

- Homogeneous
- 2. Absolute time
- Homogeneous
- Infinite
- Continuous
- Uniform
- Inert with respect to its contents
- Creation must have taken place in time, and not have been, as theology demanded, the beginning of time

- Homogeneous
- 3. Matter
- Discontinuous
- Innumerable, perfectly rigid, indivisible and indestructable atoms or 'corpuscles'
- Solidity
- Spatial extension
- Inertia
- Fills portions of space

- Discontinuous
- 4. Motion or change
- Fills portions of time
- Time is in some sort the the Space of Motion
- Nothing but motion can be either the cause, or the effect, of motion
- Motion is the only one of four entities whose existence requires all the other three: it takes place in time and space, and its vehicle is a material body

- Fills portions of time

- Pierre Gassendi (1592-1655)
- 13.3 The changed nature of mathematical proportion
- The relations governing the motion of bodies in space and time in the Newtonian mechanical universe are determined, no less than in the world of the Timaeus, by precise mathematical proportions
- Why did this new world-picture undermine the necessary and fundamental connection between natural philosophy and architectural proportion?
- The problem is that the universe now revealed by science was in fact so completely abstract and mathematical that it ceased to have any apparent relation to the world of senses
- Galileo: Having shown that many sensations which are supposed to be qualitites residing in external objects have no real existence save in us...I now say that I am inclined to believe heat to be of this character
- Furthermore, while the new equations saved the appearances - that is, there seemed to be, a perfect fit between the mathematical formulae and phenomena - they did not explain the phenomena as such
- Newton's equation for gravity did not explain how masses could act upon each other at a distance

- Newton's equation for gravity did not explain how masses could act upon each other at a distance

- The problem is that the universe now revealed by science was in fact so completely abstract and mathematical that it ceased to have any apparent relation to the world of senses

- The relations governing the motion of bodies in space and time in the Newtonian mechanical universe are determined, no less than in the world of the Timaeus, by precise mathematical proportions
- 13.4 Science and art
- The connection between Baroque architecture and the physics that coincided with its development, is not a causal connection
- The paralled between architecture and science is significant on only one condition: as evidence of a more vaguely defined 'concurrence of sensibilities', a new appreciation of and attraction towards more dynamic forms in general
- Heinrich Wolfflin identifies 5 characteristics of the transition from 'classic' to Baroque
- 1.The development from the linear to the painterly...In the former case the stress is laid on the limits of things; in the other the work tends to look limitless
- 2. The development from plane to recession. Classic art reduces the parts of a total form to a sequence of planes, the baroque emphasizes depth
- 3. The development from closed to open form
- 4. The development from multiplicity to unity. In the system of classic composition, the single parts, however firmly they may be rooted in the whole, maintain a certain independance...For the spectator, that presupposes an articulation, a progress from part to part, which is a very different operation from perception as a whole...In both styles unity is the chief aim...but in one case unity is achieved by a harmony of free parts, in the other, by a union of parts in a single theme, or by the subordination, to one unconditioned dominant, of all other elements
- 5. The absolute and the relative clarity of the subject

- 1.The development from the linear to the painterly...In the former case the stress is laid on the limits of things; in the other the work tends to look limitless
- It seems that the sixteenth and seventeenth centuries experienced a general shift of emphasis from static to dynamic space, from delimited and discrete to the infinite and continuous

- The connection between Baroque architecture and the physics that coincided with its development, is not a causal connection
- 13.5 Breaking the bond between science and art
- The reduction of artistic decisions to matters of taste
- As the objective grounds of judgement slipped away, it was natural to turn inwards: to study the mental processes of the individual artist or the individual user
- I shall argue, however, that it is still possible to construct a new objective foundation for proportion upon the ruins of the old cosmology left by the empiricist and phenomenalist world-view

- The reduction of artistic decisions to matters of taste

- Chapter 14: From the outer to the inner world
- 14.1 Clearing the ground of the obstacles to knowledge
- Newton's mathematical laws 'saved' the appearances, but they did not explain them; they did not help us understand the underlying causes
- The more the grounds of human knowledge were investigated, the less solid and knowable the 'real' world, ( as distinct from the world of appearances that could be experienced), began to appear
- Plato's distinction between the products of pure thought and the products of sensation was as relevant as ever, but the empiricist philosophers drew the opposite conclusion: that only that which could be experienced by the senses was a proper object of knowledge

- Newton's mathematical laws 'saved' the appearances, but they did not explain them; they did not help us understand the underlying causes
- 14.2 The duplication of worlds
- Locke: Every object is duplicated: in so far as it is perceived, it exists as an idea in the mind; and this perception is presumed to be caused by, but not identical with, a material object 'out there' in the external world
- Locke in his attempt to clear the philosophical ground upon which physics stands, unintentionally reveals the shakiness of its foundations
- Berkeley: These arguments which are thought manifestly to prove that colors and tastes exist only in the mind...may with equal force be brought to prove the same thing of extension, figure, and motion

- Locke: Every object is duplicated: in so far as it is perceived, it exists as an idea in the mind; and this perception is presumed to be caused by, but not identical with, a material object 'out there' in the external world
- 14.3 The dissolution of mind
- Hume questions the rational grounds for believing in the existence, not only of material bodies, but also of minds, as distinct from their contents

- Hume questions the rational grounds for believing in the existence, not only of material bodies, but also of minds, as distinct from their contents
- 14.4 Passion replaces reason
- Passion, in the case of artistic judgement, must always take the lead in creative decisions
- The analogy between art and morality precedes the idea that reason has no place in art. Reason cannot show how prefering the destruction of the world to the scratching of one's finger is wrong

- Passion, in the case of artistic judgement, must always take the lead in creative decisions
- 14.5 Berkely and Hume on proportion
- Judgments of proportion must be subject to considerations of utility
- The proportions of a chair cannot be considered beautiful if they do not make for comfort
- If you should invert a well proportioned door, the figure would be the same, but without that beauty in one situation which it had in another
- Hume's intention is clear: it is to show that aesthetic judgement, like moral one, are determined by fitness to circumstances, rather than by calculation
- The sensation fo Beauty does not lie in the geometry, but in the particular disposition of the observer

- Judgments of proportion must be subject to considerations of utility
- 14.6 Edmund Burke's attack on the theory of proportion
- Associates the sublime, which he regards as 'productive of the strongest emotion which the mind is capable of feeling', with terror, and beauty with those qualities in things which arouse a disinterested kind of love
- Beauty
- 1. Comparatively small
- 2. Smooth
- 3. Variety in the direction of parts
- 4. Parts not angular, but melted as it were into each other
- 5. Deleicate frame
- 6. Colors clear and bright, but not strong and glaring
- 7. If it should have glaring color, to have it diversified with others

- 1. Comparatively small
- Dismissed the idea that mathematical proportions are a cause of beauty in plants and animals on the grounds that they exhibit an infinite variety of proportions, yet we consider them equally beautiful
- Systems of proportion continue to be relevant even if we no longer believe that they are derived from nature's harmony, for the necessity of proportion arises precisely because we cannot find such an order in our natural environment

- Associates the sublime, which he regards as 'productive of the strongest emotion which the mind is capable of feeling', with terror, and beauty with those qualities in things which arouse a disinterested kind of love
- 14.7 The rational imperitive
- Hume: All reasoning can be divided into two kinds
- 1. Relations of Ideas
- Three times five equals the half of thirty

- Three times five equals the half of thirty
- 2. Matters of fact
- Awareness based only on induction

- Awareness based only on induction

- Reason cannot prove recurrance, nor the mechanism of causation, nor the way in which on event actually gives rise to another
- All inferences from experience are based on habit, not reasoning
- Hume's legacy is a fear of all rational speculation about matters of fact is a form of metaphysics, and therefore to be rejected
- Einstein: one is easily led to believe that all those concepts and propositions which cannot be deduced fro mthe sensory raw material are, on account of their 'metaphysical' character, to be removed from thinking

- Einstein: one is easily led to believe that all those concepts and propositions which cannot be deduced fro mthe sensory raw material are, on account of their 'metaphysical' character, to be removed from thinking
- 4 Illustrations of the priority of conjecture over observation or experience in scientific advance
- 1. Copernicus' replacement of the geocentric unverse by a heliocentric one was not necessitated by any new empirical evidene; the observations that made the Ptolemaic system untenable followed much later
- 2. One of the triumphs of Newton's system of the heavens was that it led to the discovery of Neptune in 1846. This discovery was made with the mind's eye, regions where sight itself was unable to penetrate
- 3. Newton might assert that 'Hypotheses have no place in experimental philosophy', but his whole system was nothing but a brilliant hypothesis
- 4. A crucial test for Einstein's theory was his prediction that the bending of light rays in the proximity of large masses such as the sun would be observed during a solar eclipse. The phenomenon was confirmed 6 years ofter the original prediction

- 1. Copernicus' replacement of the geocentric unverse by a heliocentric one was not necessitated by any new empirical evidene; the observations that made the Ptolemaic system untenable followed much later
- Leitmotiv for the present book: Without waiting, passively, for repetitions to impress or impose regularities upon us, we actively try to impose regularities upon the world

- 14.8 Kant: the understanding as the architect of nature
- In order to reconcile Newton with Hume, Kant had to show how we can acquire, through observation, not just a succession of provisional pictures of the world, but a final and absolutely true picture
- Critique of pure reason: Experience itself is a mode of cognition which requires understanding. Before objects are given to me, that is, a priori, I must presuppose in myself laws of the understanding which are expressed in conceptions a priori. To these conceptions, then, all the objects of experience must necessarily conform
- These fundamental components of the Newtonian system of nature are thus transplanted from the external to the internal world
- We must seek a foundation for architectural proportion in the structure of the mind

- In order to reconcile Newton with Hume, Kant had to show how we can acquire, through observation, not just a succession of provisional pictures of the world, but a final and absolutely true picture

- Chapter 15: The golden section and the golden module
- 15.1 The rediscovery of the golden section
- There is no positive evidence that the golden section, or division in extreme and mean ratio, was used as the basis for the Parthenon and other Greek temples
- P.H. Scholfield even concludes that it is not very clear how far it is correct to speak of the rediscovery of the principle of the golden section in the nineteenth century...A fairly good case could be made out for the view that the nineteenth century actually discovered the golden section as an instrument of architectural proportion, however close other periods may have come to this discovery

- There is no positive evidence that the golden section, or division in extreme and mean ratio, was used as the basis for the Parthenon and other Greek temples
- 15.2 Adolf Zeising
- The golden section is so to speak a 'goal' to which the body 'aspires', and which it achieves at maturity
- Proclaimed the golden section as the key to proportion in nature and in art

- The golden section is so to speak a 'goal' to which the body 'aspires', and which it achieves at maturity
- 15.3 Gustav Fechner
- Had he followed his own insights, and explored the mathematical and aesthetic consequences of his pantheistic geometry of nature, Fechner might have re-established the study of proportion on the ancient empathic vision of cosmic unity
- But the world, writes E.G. Boring, 'chose for him; it seized upon the psychological experiments, which Fechner meant merely as contributory to his philosophy, and made them into an experimental psychology

- Had he followed his own insights, and explored the mathematical and aesthetic consequences of his pantheistic geometry of nature, Fechner might have re-established the study of proportion on the ancient empathic vision of cosmic unity
- 15.4 The statstical approach to the golden section
- Beauty had to be examined as a purely psychological phenomenon
- Fecher used 3 kinds of experimental approach
- 1. Choice-subjects select or reject certain proportions
- 2. Construction-subjects complete a figure
- 3. Use-involving everyday objects

- 1. Choice-subjects select or reject certain proportions
- Most popular shape was the golden section, but 65% did not chose the golden section

- Beauty had to be examined as a purely psychological phenomenon
- 15.5 Attempts to explain the phenomenon
- The Golden ratio provided the ideal balance of unity and variety that was unconsciously perceived
- However, M. Borissavlievitch rejects the notion that the eye can perceive two ratios simultaneously

- However, M. Borissavlievitch rejects the notion that the eye can perceive two ratios simultaneously
- M. Borissavlievitch says the golden section represents the balance between two unequal asymmetrical parts, which means that the dominant si neither too bid nor too small
- Statistical method reduces art and proportion to its most banal manifestation: mass appeal

- 15.6 The curves of life
- Theodore Cook criticises those who would limit the role of science to the description of observations, without speculation about the first causes and ultimate ends
- Newton arrived at his theory of the movements of the celestial bodies in our own solar system by postulating perfect movement and by calculating from that the apparently erratic orbits of the planets. In jsut he same way may it not be possible to calculate and define the apparently erratic growth and forms of living things?

- Theodore Cook criticises those who would limit the role of science to the description of observations, without speculation about the first causes and ultimate ends
- 15.7 The golden section in Le Corbusier's early work
- The climate of thought Corb was absorbed in was permeated by proportional ideas
- Le Corbusier saw the connection between standardization and systems of proportion
- Has not the car been perfeted by its standardization, and was not the Greek temple perfected in the same way?
- Regulating lines
- His prewar villas illustrate the long development that finally led him to the modular

- The climate of thought Corb was absorbed in was permeated by proportional ideas
- 15.8 Origins and aims of the modular
- Modular was a complete system of measures that automatically generated golden sections and related ratios in whatever he designed
- Post-war standardization was an issue answered by Corb with a system of measures and proportions based on the human body
- Corb aimed to devise an efficient method for rationalizing construction and rebuilding after the war
- Le Corbusier assumes 2 things
- 1. The golden section holds the definitive key to the proportions of the human body
- 2. The godlen section can be constructed geometrically by finding the place of the right angle slightly off center within a double square

- 1. The golden section holds the definitive key to the proportions of the human body
- Le Corbusier was wrong on both assumptions

- Modular was a complete system of measures that automatically generated golden sections and related ratios in whatever he designed
- 15.9 The geometry of the modular
- The first principle of the modular was thus flawed: the 'place of the right angle' is only an approximation
- 6 thousandths of a value are negligible in construction, but are infinitely precious in philosophy

- The first principle of the modular was thus flawed: the 'place of the right angle' is only an approximation
- 15.10 The modular's arithmetic
- All dimensions must be capable of being measured out, and therefore commensurable
- The sum of two consecutive measures must equal the next if measurements are to fit together exactly in a building
- For a proportion system to be effective as a means of knitting together a whole composition, the eye of the observer must be led by a series of steps from the smaller to larger units
- Le Corbusier's attempts to reconcile the metric and imperial systems

- All dimensions must be capable of being measured out, and therefore commensurable
- 15.11 The anthropometrics of the modular
- The body does not bend at the solar plexus or at the navel, wherr Le Corbusier shows it cut by the golden section, but at the hip joint
- The criticism of the modular is that it sets up a single rigid set of proportions, instead of recognizing proportionality as such
- The modular lacks some of the most necessary functional dimensions, such as the height of a normal door or the length of a bed
- The conclusion is that Le Corbusier, vitruvius, Leonardo, imposes a rational mathematical schema upon a generalized image of the body

- The body does not bend at the solar plexus or at the navel, wherr Le Corbusier shows it cut by the golden section, but at the hip joint
- 15.12 The modular in practice: the Unite de Habitation
- Major example of Le Corbusier's use of modular
- Three shapes composing the overall building
- 1. Long Elevations: very nearly a golden rectangle
- 2. End elevations: a root 5 rectangle
- 3. Plan: a golden root 5 rectangle: that is, three transverse golden rectangles alternating with four quares, a ratio of 5.854:1

- 1. Long Elevations: very nearly a golden rectangle
- Le Corbusier seems content to select measures from the scale more or less arbitrarily, purely on the basis of whether or not the chosen measure is close enough to the functionally required dimensions
- Proportions must be designed to be as clear as possible, and are extended to encompass the scheme as a whole
- Le Corubsier's failure to resolve the conflict in his own thinking between 'empathy' and 'abstraction', was the origination of the modular's shortcomings

- Major example of Le Corbusier's use of modular

- Chapter 16: The house as a frame for living and a discipline for thought
- 16.1 Architecture: a practical or speculative art?
- Even Kant himself saw that his definition of the aesthetic object as 'disinterested' and 'purposeful without purpose' was inapplicable to architecture
- Seperation of architecture from building
- An abstract architecture - an architecture of pure proportion is possible

- Even Kant himself saw that his definition of the aesthetic object as 'disinterested' and 'purposeful without purpose' was inapplicable to architecture
- 16.2 Physical versus intellectual functions
- Ruskin reflects the post-enlightenment seperation of form and use, architecture and construction
- Van der Laan says architecture is building, but not in a Functionalist manner
- Adolf Loos objects to the profanation of sacred things by misappropriating them for material use
- Le Corbusier makes a division between work and thought, or work and meditation
- In Towards a New Architecture, Corbusier unites both ideas-the masking of construction and the non utilitarian nature of architecture

- Ruskin reflects the post-enlightenment seperation of form and use, architecture and construction
- 16.3 The abstract revolution
- Whatever validity the seperation from construction may have had in the mid-nineteenth century it has certainly lost by the late twentieth
- It was realised by Le Corbusier and others that architecture can only be defined by what unites all building, not by what divides one building from another
- The essential property of building as such
- Marc-Antoine Laugier and the primitive hut

- Whatever validity the seperation from construction may have had in the mid-nineteenth century it has certainly lost by the late twentieth
- 16.4 The necessary and the essential
- Laugier's cruical discovery was to distinguish the essential from the necessary
- Necessary: Baths, hot water, kitchen
- Essential: Four cylindrical uprights and a roof

- Necessary: Baths, hot water, kitchen
- Demarcation of habitable space

- 16.5 The search for a starting point
- The common core of building must embody a form
- Laugier's statement: architecture's principles are to be found on nature itself, and in nature's processes are to be found clearly written the laws of architecture
- Art does not imitate nature, but nature, art
- The functional programme, which Summerson rightly identifies as the 'one new principle' contained in Modernist theory, is really just 'nature' in a new dress: a force outside architecture, which determiens what architecture shall be
- We must satisfy the need for an intrinsically architectonic generator of form

- The common core of building must embody a form
- 16.6 The intrinsic meaning of architecture
- Van Der Laan's body:house:world establishes a proportional relation
- Architecture as such can express only a single idea: the elementary principle of human habitation, the demarcation by walls of a measured habitable space within the measureless space of nature
- Architecture expresses the building's essential relation to the world

- Van Der Laan's body:house:world establishes a proportional relation
- 16.7 The first building
- Van Der Laan: The essential nature of the architectonic act, the act of building, is the placing of solid elements on the ground in such a way that they mark out a space between them
- The relation of the space to the walls is not a question of form, however, but of proportion
- This then is the first building: just two parallel walls set a determined distance apart

- Van Der Laan: The essential nature of the architectonic act, the act of building, is the placing of solid elements on the ground in such a way that they mark out a space between them
- 16.8 Articulate building
- Van Der Laan: Homogeneous, unarticulated huilding-forms can never lay claim to architectonic expressiveness; there must always be a whole and parts
- The core of architecture is con-struction, the putting together of seperate parts to make an articulate whole
- This articulation involves relations between parts, and therefore an embryonic notion of scale or proportion
- We are concerned with two kinds of relations
- 1. The relation within each thing between its seperate parts, and between these and these and the whole
- Quantifiable

- Quantifiable
- 2. The relation of the artefact to other domains-for instance to the natural world or the world of abstract ideas
- Based on analogy

- Based on analogy

- Van der Laan devised a typology of twenty-five arrangements of three forms
- Pure proportion sets the process of articulation going

- Van Der Laan: Homogeneous, unarticulated huilding-forms can never lay claim to architectonic expressiveness; there must always be a whole and parts
- 16.9 What system of proportion do we need?
- Vitruvius
- Ordonnance
- Balanced agreement of the measures of the building's measures...This is achieved through quantity

- Balanced agreement of the measures of the building's measures...This is achieved through quantity
- Symmetry
- Proper mutual agreement between the members of the building, in accordance with a certain part selected as standard, of the seperate part to the figure of the building as a whole

- Proper mutual agreement between the members of the building, in accordance with a certain part selected as standard, of the seperate part to the figure of the building as a whole
- Proportion
- Correspondence among the measures of the members of an entire work, and of the whole to a certain part selected as standard

- Correspondence among the measures of the members of an entire work, and of the whole to a certain part selected as standard

- All three definitions make the same three demands
- 1. The measures of all the parts and of the whole must agree or correspond with each other. I take this to mean that they must constitute a chain of related measures, perhaps connected by a constant ratio between one measure and the next. In that case, they would form a geometric progression
- 2. There must be a direct relation between the whole or largest measure and an elementary unit or standard module, derived from the smallest part: for example, in Classical architecture, the column diameter
- 3. It follows from 1. and 2. that the measures of all intermediate parts must relate to the whole and to the unit

- 1. The measures of all the parts and of the whole must agree or correspond with each other. I take this to mean that they must constitute a chain of related measures, perhaps connected by a constant ratio between one measure and the next. In that case, they would form a geometric progression
- The plastic number has a strong claim to meeting all three of Vitruvius' demands

- 16.10 Knowing by not knowing
- Hume's philosophy undermines the very foundation of science since we cannot be certain of nature; matters of fact vs. pure reason
- Kant meets Hume's challenge head on by stating that nature is intelligible because the understanding itself is the lawgiver of nature
- Van Der Laan treats the problem of architectural proportion in a similar way to Kant

- Kant meets Hume's challenge head on by stating that nature is intelligible because the understanding itself is the lawgiver of nature
- By making things, we make our world, and to the extent that we have made it, we can know it
- I cannot give a name to a concrete size in the way that I can give a name to a number, one, two, three, and so on. But I can indeed give a name to things that are of roughly the same size: I call them 'of the same size'. Now it is no longer a question of the size of a given thing, but of a thing of a given size: a size of which I can form a mental concept
- Van Der Laan's theory of proportion, and indeed of architecture, might be reduced to this: that there are limits within which sizes can be related to each other, and other limits beyond which this relation breaks down. Taking these limits as a basis, it is possible to establish a chain of relationships by which the whole architectonic environment, from a brick to the whole town, can be connected together and made intelligible and humane.
- A piece of the unknown has become known: and precisely by exploiting the fact that we cannot know it

- A piece of the unknown has become known: and precisely by exploiting the fact that we cannot know it

- 16.11 Types and orders of size
- 1. Within the limits of a type of size we call all concrete measures identical; there is as yet no question of proportion
- 2. Within the limits of an order of size the types of size can be compared with each other; here it is a question of proportion
- 3. Beyond the limits of an order of size no relation is any more possible between types of size; there can no longer be any question of porportion

- 1. Within the limits of a type of size we call all concrete measures identical; there is as yet no question of proportion
- 16.12 Conclusion
- The mathematics of architectural proportion are necessarily very simple
- A disocvery in architecture or science can take centuries to make its influence felt on the other
- Design of Greek temples led to developments in ancient mathematics
- Gothic cathedral foreshadowed the Scholastic summa
- Mannerist art pre-figured certain aspects of 17th century science

- Design of Greek temples led to developments in ancient mathematics
- Necessary to embody a mathematical order in our works of art
- Sigfried Giedion's whole argument nevertheless relies on a quasi-mystical causative connection: the spirit of the age is at work behind the scenes
- Architecture simply 'gives measure' to our spatial environment

- The mathematics of architectural proportion are necessarily very simple