1.1 A copy of a diagram in Calcidius’ On Plato’s Timaeus on the left page combined with a Lambda diagram representing the harmonic structure of the World Soul on the right page. From a copy of Macrobius’ Commentary on the Dream of Scipio (ca. 1175–1200), fols. 65v, and 66r in the collection of the Walters Art Museum 9
1.2 Renaissance engraving showing Apollo, the Muses, the planetary spheres and the musical modes. From Franchinus Gaffurius’ Practica musice (1496) 22
1.3 The human being regarded as a universe in miniature. Line engraving by T. de Bry, 1617. Title page of Robert Fludd, Utriusque Cosmi … Historia 30
1.4 Francesco di Giorgio Martini (1439–1501), design for a church based on the concept of musica humana. Drawing from the Codex Magliabechiano, Biblioteca Nazionale, Florence 40
2.1 Equilateral and scalene right-angled triangles 68
2.2 Cube composed of isosceles right-angled triangles 69
2.3 Two geometrical proportions of four terms each: 1, 2, 4, 8 and 1, 3, 9, 27. From Marsilio Ficino, Opera Omnia, vol. II, 1459 72
3.1 Arrangement of six and twelve as rectangular shapes 80
3.2 Geometrical explanation of the movement of the Sun according to the model of the eccentric circle 91
3.3 Model showing the circular motions of the Sun 92
3.4 Table illustrating the harmonic structure of the Soul 96
6.1 Three diagrams illustrating the harmonics of the Timaeus. The first is an illustration of the Greek letter lambda representing the harmonic numbers in the World Soul (taken from Calcidius, On Plato’s Timaeus, ed. Magee, diagram 7 (chapter 33, 47, 95), 696, the second is diagram 8 (chapters 41, 47), 696 and the third is diagram 9 (chapter 48), 697 169
6.2 The Timaean scale based on the numbers in the World Soul 170
6.3 Macrobius, musica mundana: the universe, the earth in the centre, surrounded by the seven planets within the zodiacal signs. From Macrobius, Commentary on the Dream of Scipio. Diagram in folio 25 recto. Source: Copenhagen, Det Kongelige Bibliotek, ms. NKS 218 4 ° 171
6.4 Full-page lambdoid diagram in Madrid Madrid, Biblioteca Nacional, MS Vitr. 14–3, f. 27r. Reproduction from Michel Huglo’s transcription in ‘Musica ex numeris’, 113 180
6.5 Woodcut showing Pythagoras with bells, a kind of glass harmonica, a monochord and (organ) pipes in Pythagorean tuning. From Franchinus Gaffurius, Theorica musicae (1492) 184
6.6 Title page with organ. From Franchinus Gaffurius, Theorica musicae (1492) 188
8.1 Pyramid Text 305.474a. From James P. Allen, A New Concordance of the Pyramid Texts. Vol. I: Introduction, Occurrences, Transcription (Providence, RI: Brown University, 2013) 232
8.2 Eleventh-century representation of John of Damascus and his foster-brother, St Cosmas of Maiuma, writing their books. From the Menologion of Basil II, ca. 1000 CE. Rome, Vatican Library, Vat. Gr. 1813 235
8.3 A group of seven famous physicians, shown in a folio from the ‘Vienna Dioscorides’, Vienna, Austrian National Library, Codex Vindobonensis Med. Gr. 1, fol. 3v 243
9.1 Early tenth-century manuscript of Calcidius’ Timaeus translation and commentary. Rome, Vatican Library. On the left, the Sun, Moon, and earth; on the upper right, a lunar eclipse with spherical bodies and a cylindrical shadow; on the lower right, the Sun with the zodiacal circle, Venus as Hesperus (in the evening), and Venus as Lucifer (in the morning). See Calcidius, Timaeus, trans. Magee, 702 and 705 257
9.2 Armillary sphere with ecliptic and zodiac. Woodcut from Christopher Clarius, In sphaerum Ioannis de Sacro Bosco commentarius, 1570 (Commentary on the De sphaera mundi of Johannes de Sacrobosco, ca. 1230, University of Paris) 261
9.3 Durham Cathedral, vaulting of the nave. Author’s photograph 265
9.4 Durham Cathedral, interior. Author’s photograph 266
9.5 Durham Cathedral, arches and ribs in the choir aisles. Author’s photograph 273
9.6 Durham Cathedral, detail of the choir vault. Author’s photograph 274
9.7 Durham Cathedral, vaulting of the nave, detail. Author’s photograph 276
9.8 Lincoln Cathedral, vaulting of the nave. Author’s photograph 279
10.1 Ficino’s model of the World Soul’s composition in terms of harmonic proportions. From Marsilio Ficino, Opera Omnia, vol. II, 1460 297
10.2 Ficino’s model of the Soul’s composition in terms of the larger numbers he found in sources such as Calcidius’ Timaeus commentary. From Marsilio Ficino, Opera Omnia, vol. II, 1461 298
10.3 A reconstruction of Ficino’s theory of the proportions between the different faculties of the human soul. Author’s drawing 305
10.4 The diapason (octave; 2:1) closing full in Man. Robert Fludd, Utriusque cosmi historia, 275 310
14.1 Plan of the Piazza della Signoria, Florence, showing its simplified geometrical arrangement, based on the rotational square (X & Y), according to Trachtenberg. Redrawn by Stephen Calcutt 395
14.2 Volumetric projection of the Piazza della Signoria, indicating cuboid volume of the principal square, in relation to the height of the campanile of the Palazzo Vecchio, after Trachtenberg. Redrawn by Stephen Calcutt 396
14.3 Ground floor plan of the Palazzo Medici, Florence, showing a fairly regular geometrical layout with central cortile surrounded by state rooms or offices. Drawing by Manwinder Lall 403
14.4 View of the Palazzo Medici, indicating cubic massing of the building as made visible by its corner location with large overhanging eaves 404
14.5 Reconstruction of the plan and section of the Rotunda of S. Maria degli Angeli, Florence, according to Eugenio Battisti. Redrawn by Manwinder Lall 407
14.6 View of the Rotunda of Santa Maria degli Angeli, from the Codex Rustici, c.1447. Florence, Library of the Seminario Arcivescovile Maggiore 408
14.7 ‘Distance-point construction’. The final step in Alberti’s account entails drawing diagonal lines across the pavement to demonstrate the ‘correct’ foreshortening of the geometry. Drawing by Manwinder Lall, after Joan Gadol 412
14.8 Sebastiano Serlio (1475–1554). Perspective construction of a four-way arch (‘Janus Quadrifrons’), from Sebastiani Serlii Bononiensis de architectura libri quinque … (Venetiis: Apud Franciscum de Franciscis Senensem, & Joannem Chriegher, 1569), ‘Liber Secundus’, p. 35, RIBA Collections 417