1. The Order of Proportion

2. ‘ For the harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty. ’ D ’ Arcy Wentworth Thomson, 1917 God as the Architect of the Universe

3. Le Corbusier: Le Modulor, 1949 ‘Nature is ruled by mathematics and the masterpieces of art… express the laws of nature, and themselves proceed from those laws.’

4. He begins with the rigid line, which is essentially abstract and alien to life …. He seeks further geometrical possibilities of line, creates triangles, squares, circles, places similarities together, discovers the advantages of regularity, in short, creates a primitive ornament which provides him not only with a mere delight in decoration and play, but with a table of symbolic absolute values, and therefore with the appeasemenet of his condition of deep spiritual distress. ’ Wilhelm Worringer, Form in Gothic, 1912

5. ‘ The naturalistic art of empathy proves to owe far more to the human, and abstract art far more to nature, than the associations of “ naturalism ” and “ abstraction ” lead us to assume. Inside every empathics there is an abstractionist fighting to get out, and vice-versa. ’ Richard Padovan, Proportion, 1999

6. ‘ Irrespective of what may be the current scientific view of this matter, this is still a fairly accurate picture of our everyday experience, and – since everyday experience is what matters for architecture – of the relationship between measure and architectural proportion. The unit is conceived as something chosen – but not chosen at random. It must be large enough to count with respect to the thing to be measured, while on the other hand, if the unit is to be of any use as a measure, it naturally cannot be of a similar size to the thing to be measured. In relation to the thing, the unit must be “ not too little and not too big. ” Padovan,

7. Geometric progression; adding each number to the preceding term. Addition of any term to any other becomes too complex. ‘ True order being a balanced combination of unity and complexity; an effective system of proportions must combine the multiplicativeness of the geometric progression with the additiveness of the arithmetic progression. ’ Padovan p.44

8. Square root of 2 1.4142 Square root of 3 1.732 Square root of 4 2.0 Square root of 5 2.236 Golden section 1.618 Plastic number ratio 1.325 ‘ Interwoven grid ’ (Pythagorean-Platonic ’ system: 1 y y 2 y 3 x xy xy 2 x y 3 x 2 x 2 y x 2 y 2 x 2 y 3 x 3 x 3 y x 3 y 2 x 3 y 3

10. The fundamental unity of geometry and number: ‘ We must finally construct (6) a second unit square ABMN adjacent to the first. The purpose of this is to reveal the additional lengths EN, GN, IN, and KN, and thus the proportionally all-important complementary numbers 1 + √2, 1 + √3, 1 +√4, 1+√4 and 1 + √5, or approximately 2.414, 2.723, 3.0 and 3.236. In this way, we provide each of the four square systems with the weft series it needs to become completely effective. Moreover, bisecting the 1 + √5 rectangle (7) we obtain two Golden Section rectangles, since Golden Section = (1+√5)/2

11. The Parthenon, Athens

12. Simple whole numbers? ‘ Egyptian triangles ’ base=8 height=5 sides=√41, just over 6.4 What determines the positioning of the underside of the entablature?

13. The Parthenon as Whole number relationships Core rectangle of 15x6 standard column spaces. Stylobate of 9:4

14. ‘ The human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is the same …’ Vitruvius c.25BC

15. Leonardo da Vinci ‘ Vitruvian Man ’ c.1500

16. Chartres Cathedral 1194-1224

18. ‘The total length of the nave and choir... comprises seven double bay units, marked out by the alternation of round and octagonal piers … We have found a series of whole number foot dimensions that would have allowed the work to be measured out with a yardstick - perhaps literally a three foot measure, since all the vertical dimensions are multiples of three feet. Cords and pegs would still have been needed… as they are today, to determine right angles.’ Padovan p.196

19. ‘For us, the outline is a certain correspondence between the lines that define the dimensions; one dimension being length, another breadth, and the third height … I affirm again with Pythagoras: it is absolutely certain that Nature is wholly consistent … The very same numbers that cause sounds to have concinnnitas, pleasing to the ears, can also fill the eyes and mind with wondrous delight. From musicians therefore … or from those objects in which Nature has displayed some evidence and noble quality, the whole method of outlining is derived’ Leon Battista Alberti (1404-72), On the Art of Building in Ten Books

20. Andrea Palladio 1508-80

21. Villa Pisani, Bagnolo, from Andrea Palladio, Four Books on Architecture

22. The Veneto villa as working farm and place of leisure

23. Villa Pisani, Bagnolo

24. Villa Pisani, Bagnolo - rear elevation

25. Villa Pisani, Bagnolo Central Hall

26. Typical windows of London houses, 18th century from Dan Cruickshank and Peter Wyld, London, the art of Georgian Building, 1975

27. Doorway from Gentleman and Builder’s Repository, by Edward Hoppus, 1737

28. Proportions of typical London house, 18th century, from Cruickshank and Wyld