Dr. Kim H. Veltman

Notes

Introduction
Ch 1. Definitions and Origins
Ch 2. Centres
Ch 3. Treatises
Ch 4. Classification
Ch 5. Instrumentation and Science
Ch 6. Art and Representation
Ch 7. Architecture and Environment
Ch 8. Imagination and Freedom
Ch 9. Epilogue

1. William M. Ivins Jr. Art and geometry, Cambridge Mass.: Harvard University Press, 1946; Martin Foss, The idea of perfection in the western world, Lincoln: University of Nebraska Press, 1946.

2. Julius von Schlosser, Die Kunstlitteratur, Vienna: Schroll, 1924; translated by Otto Kurz as La letteratura artistica, Scandicci (Florence): La nuova Italia editrice, 1964 (reprint 1986).

3. Samuel Y. Edgerton, Jr., The Renaissance Rediscovery of Linear Perspective, New York: Basic Books, 1975.

Chapter 1

1. Objective as used here refers to a systematic relationship between object and representation without necessary intervention of the human eye. A school of modern artists characterized by Hansen (e.g. 1987) disagrees with this definition and would include as objective all mathematical systems which attempt to record impressions of or at the eye.

2. Vitruvius, The Ten Books on Architecture, tr. Morris Hicky Morgan, New York: Dover, 1960, p. 198. For the original Latin see, Vitruvii De architectura libri decem, ed. Dr. C. Fensterbusch, Darmstadt: Wissenschaftliche Buchgesellschaft, 1976, p. 308.

3. See, for instance, John White, The Birth and Rebirth of Pictorial Space, London: Faber and Faber, 1957, chapter XVI, "Spatial design in Antiquity." The other major exponent of the view that the Greeks had knowledge of linear perspective is A.M.G. Little, Roman Perspective Painting and the Ancient Stage, Shiremanston: Moretus Press, 1971. Against this view is G.M.A. Richter, Perspective in Greek and Roman Art, London: Phaidon, 1970.

4. Vitruvius, tr. Morgan, as in note 2, p. 211; c.f. Vitruvius, ed. Fensterbusch, pp. 332-335.

5. See, for instance, H .G. Beyen, Die Pompejanische Wandekoration vom und bis zum 4 Stil, The Hague: Martinus Nijhoff, 1938-1960, 2 vols.

6. See, Alfonso de Franciscis, La villa romana di Oplontis, Recklinghausen: Verlag Aurel Bongers, 1975.

7. This point has been made by Miriam Shild Bunim, Space in Mediaeval Painting and the Forerunners of Perspective, New York: Columbia University Press, 1940, p. 31.

8. John Pennethorne, The Geometry and Optics of Ancient Architecture, London: Williams and Norgate, 1878.

9. Plato, "The Sophist," 236 a-c in The Collected Dialogues of Plato ed. Edith Hamilton and Huntington Cairns, Princeton: Princeton University Press 1961; p. 978-979 (Bollingen Series LXXI).

10. For this and other examples see Jurgis Baltrusaitis, Anamorphoses, ou magie artificielle des effets merveilleux, Paris: Olivier Perrin, 1969, pp. 10-14. There is an English translation by W.J. Strachan, Anamorphic Art, Cambridge: Chadwyck-Healey Ltd., 1977.

11. For an introduction to this complex story see J. Baltrusaitis, as in note 10 above, pp. 71-78. For more details see Hans Willem Van Helsdingen, Historier en peindre: Poussins opvattingen over kunst in het licht van de discussies in de franse kunstlitteratur in de twede helft van de zeventiende eeuw, Proefschrift, Utrecht, 1971.

12. L. F. Shegin, Die Sprache des Bildes, Form und Konvention in der alten Kunst, Dresden: VEB Verlag der Kunst, 1982.

13. This point was made in Leon Battista Alberti, On Painting, ed. John Spencer, New Haven: Yale University Press, 1956, p. 121. Cf. Alberti, De pictura ed. Cecil Grayson, Rome: Laterza, p. 55: "id velum quod ipse inter familiares meos sum solitus appelare intercisionem, cuius ego usum nunc primum adinveni."

14. For a transcription of this passage see the author's Leonardo da Vinci and the Visual Dimensions of Science and Art (Munich: Deutscher Kunstverlag, 1986), p.443.

15. See Michel Gallet, Les origines de la perspective linéaire. Evolution de l'optique picturale pendant la renaissance. Thèse presentée pour le diplôme supérieur de l'Ecole du Louvre, 1955.

16. Joseph Harnest, "Dürer und die Perspektive" in: Albrecht Dürer, ed. Peter Strieder, Königstein: Langewiesche, Karl Robert, Nachfolger Hans Koster, 1981, p. 358.

17. Our figure 17 is a translation from the classic article by M. D'Avezac "Coup d'oeil historique sur la projection des cartes de géographie," Bulletin de la société de géographie, Paris, avril, mai, juin 1863, p. 152.

For a brief introduction to historical methods see Johannes Keuning, "The history of geographical map projections until 1600," Imago mundi, London, vol. 12, 1955, pp. 1-24. For a standard modern textbook on the subject see J.A. Steers, An Introduction to the Study of Map Projections, London: University of London Press Ltd., 1927, etc. On the complex problems of Ptolemy's cartographic methods see: Mollweide, "Mappirungkunst des Claudius Ptolemaeus, ein Beitrag zur Geschichte des Landkarten," Monatliche Correspondenz zur Beforderung der Erd und Himmelskunde von Baron von Zach, Gotha, Bd. 11, Apr. 1805, S. 319-340, 504-415; "Die Gradnetze des Ptolemaus im ersten Buche seiner Geographie. šbersetzung der Kapitel 21 bis 24 nebst Anmerkurgen und Figuren," Beilage zum Jahresbericht des Königlichen Gymnasiums zu Chemnitz fur das Schuljahr, Ostern 1908, bis Ostern 1909....Chemnitz: Druck von J.C.F. Pickenhahn & Sohn, pp. 1-28 (Programm-Nr. 726). The idea that Ptolemy developed a third method of projection was put forward by Konstantin Cebrian, Geschichte der Kartographie. I. Altertum, Gotha; Justus Perthes, 1922, p. 98 and Abb. 6 and developed by Otto Neugebauer, "Ptolemy's Geography. Book VII, Chapter 6 and 7", Isis, Cambridge, Mass., vol. 50, 1959, pp. 22-29; Kirsti Andersen, "The Central projection in one of Ptolemy's map projections," Centaurus, vol. 30, 1987, pp. 106-113. Also important in this context are Lauri O. Th. Tudeer, "On the origin of the maps attached to Ptolemy's Geography," Journal of Hellenic Studies, London, vol. 37, 1917, pp. 62-76; Dr. Paul Schnabel, "Die Enstehungsgeschichte des kartographischen Erdbildes des Klaudios Ptolemaios," Sitzungsberichte der preussischen Akademie der Wissenschaften, Philosophisch-historische Klasse, Berlin, Stuck VIII, 6 März 1930, 214-250; Hans von Mzik, "Neue Gesichtspunkte zur Würdigung der Bedeutung der Geographie des Klaudios Ptolemaos fur die Orientalistik...," Litterae orientales, Leipzig, vol. 54, Apr. 1933, 1-16; Ptolemaeus, Theorie und Grundlagen der Darstellenden Erdkunde, tr. Hans v. Mzik, Friedrich Hopfner, Vienna: Gerold und Co., 1938. (Klotho, Bd. 5. Des Klaudios Ptolemaios Einführung in die darstellende Erdkunde); Leo Bagrow, "The origin of Ptolemy's Geographia," Geografaska annaler. Utgivna av Svenska sallskapet for antropologi och geografi, Stockholm, vol. 27, 1945, pp. 318-387.

It is perhaps not irrelevant to note that Danti (1583) in his annotation to Problema VIII. Proposizione XXXVI refers not to Ptolemy's Geography but rather to his Almagest, Bk. I, Chap. 9 regarding inscription of regular polygons in circles and similar geometrical problems. Stevin (1605) in his 1st definition mentions that: "Optics as genus has several species such as catoptrics, dioptrics, planispheres, sun-dials, perspective proper and several others." In other words there are links between optics, perspective and astronomy rather than with geography.

18. The relevant passages have been translated and analysed in the author's work on Leonardo, as in note 14 above, pp. 143-169

19. See, for instance, Timothy K. Kitao, "Prejudice in perspective: a study of Vignola's perspective treatise," Art Bulletin, New York, vol. 44, Sept. 1962, pp. 173-194 and Robert Klein, "Pomponius Gauricus on perspective," Art Bulletin, New York, vol. 43, Sept. 1961, pp. 211-230.

20. Cf. Wilfred Theisen, The Mediaeval Tradition of Euclid's Optics, University of Wisconsin, Ph.D. Thesis, 1972.

21. Al-Farabi, Catalogo de las ciencias, ed. A. G. Palencia, Madrid: Consejo superior de investigaciones cientificas, Patronato Menendez y Pelayo, Instituto Miguel Asia, 1953.

22. See the author's thesis Renaissance Optics and Perspective. A Study in the Problems of Size and Distance, D. Phil. Thesis, Warburg Institute, University of London, 1975, pp. 15-73.

23. Spencer edition, as in note 13 above, p. 69.

24. See Rocco Sinisgalli, Borromini a quattro dimensioni, Rome: Universita degli studi di Roma - Facolta di architettura, 1981.

25. Cf. the title to Vredeman de Vries, Perspective, id est, celeberrima ars inspicientis aut transpicientis oculorum aciei in pariete...Leiden: Hondius 1604.

26. For a standard analysis of Saenredam's methods see Rob Ruurs, Saenredam. The Art of Perspective, Amsterdam: Benjamins/Forsten Publisher, 1987. (Oculi, Studies in the arts of the Low Countries, volume 1.)

27. See: Il libro di Victruvio architecto tradocto di latino in lingua et sermone proprio et volgare da Fabio Calvio Ravenate. In casa di Raphaello da Giovanni di Sacte da Urbino et a sua instantia, Munich, Bayerische Staatsbibliothek MS. 1034, Ital. 37 inter cimel. 380.

For the context of this and other Vitruvian editions see Pier Nicola Pagliari, "Vitruvio da testo a canone" in Salvatore Settis, ed. Memoria dell antico nell'arte italiana, Turin: Giulio Einaudi, 1986, pp. 7-85, which is followed by an article by Arnold Nesselrath, "I libri di disegni di antichita. Tentative di una tipologia," pp. 89-147 and a basic contribution by Settis himself: "Continuita, distanza, conoscenza. Tre usi dell'antico," pp. 375-486.

28. Views of Rome then and now. 41 etchings by Giovanni Battista Piranesi and corresponding photographs and text by Herschel Levit, New York: Dover Publications, 1976.

29. Edgerton, as in note 3 of the introduction, pp. 74-75.

30. Sir E. H. Gombrich, Means and Ends. Reflections on the History of Fresco Painting, London: Thames and Hudson, 1976, p. 32.

31. See Antonio Manetti, The Life of Brunelleschi, trans. Catherine Enggars, ed. Howard Saalman, University Park: Pennsylvania State University Press, 1970, pp. 118, 151.

Some of these connections have been considered in the author's Military Surveying and Topography: the Practical Dimension of Renaissance Linear Perspective. Lisbon: Junta de investigacoes cientificas da ultramar 1979. (Centro de estudos de cartografia antiga. CXXIX.)

32. See: Jean Lejeune, Les van Eyck. Peintres de Liège et de sa cathédrale, Liege: Georges Thone, 1956; Armand Jean Heins, Une rue de Gend peinte par Hubert van Eyck. Essai d'identification de la vue de ville représentée sur le revers de deux volets de polyptique de l'Agneau mystique, Ghent: N. Heins, 1907.

33. Cord Meckseper, Kleine Kunstgeschichte der deutschen Stadt in Mittelalter, Darmstadt: Wissenschaftliche Buchgesellschaft, 1982. Cf. Pierre Lavedan, Représentation des villes dans l'art du moyen age, Paris: Vanoest, 1954 and Claudio Buttafava, Visioni de citta nelle opere d'arte del medioevo e del rinascimento, Milan: Libreria Salto, 1963. (Politecnico di Milano, Istituto di urbanistica della facolta di architettura.)

34. Cf. IR. R. Meischke, Amsterdam Burgerweeshuis, Hague: Staatsuitgeeverig, 1975 (De nederlandse monumenten van geschiedenis en Kunst. De Provincie Noordholland. De gemeente Amsterdam Deel 1)

35. Giovanni Paolo Lomazzo, Idea del tempio della pittura, Milan: P.G. Pontio, 1590, p. 17: Mantegna c'ha fatto alcuni disegni di prospettiva, dove ha delineato le figure poste secondo il suo occhio, delle quali io ne ho veduto alcune di sua mano con suoi avertimenti in scritto.

36. Ibid...pp. 17, 52, 149. Cf. Giovanni Paolo Lomazzo, Trattato dell'arte de la pittura, Milan: Appresso Paolo Gottardo, 1585, pp. 336, 100-101.

37. Benvenuto Cellini, Trattati dell'oreficeria e della scultura, ed. C. Milanesi, Florence: Le Monnier, 1857, p. 226:

un libro scritto in penna... In fra le altre mirabili cose, ch'erano in su esso, trovai un discorso della prospettiva, il piu bello che mai fusse trovato da altro uomo al mondo, perche le regole della prospettiva mostrano solamente lo scortare della longitudine e non quella della latitudine a altitudine. Il detto Leonardo aveva trovato le regole re le dava ad intendere con tanta bella facilta et ordine, chi ogni uomo che la vedeva era capacissimo.

38. Lomazzo Re: Bramante see; Lomazzo, 1585, as in note 36, pp. 370, 100, 320 and 1590, as in note 35, pp. 150, 16. Re: Bramantino see: Lomazzo, 1585, 271-272, 274-275, Lomazzo, 1590, 150, 16, Re: Foppa see: Lomazzo, 1585, 275, 100, 320 and Lomazzo 1590, 108, 150.

39. Luca Pacioli, Libellus in tres partiales tractatus in Divina proportione, Venice, 1509, fol. 22r:

Lectore non te maravigliare se de simili corpi composti de diverse e varie base non te se mette sempre in margine loro figure conciosia ch'le sieno facte per mano de bono perspectivo que non si possano sempre havere a sua posta come sua humanita feci el nostra Lionardo da vinci.

Chapter 2

1. Giorgio Vasari, Lives of the Painters, Sculptors and Architects, trans. William Gaunt, London: Dent, 1927 (Everyman's Library 784-787), 4 vol.

2. Jacob Burckhardt, Die Cultur der Renaissance in Italien, Basel: Schweighauser, 1860. Trans. Dr. Ludwig Geiger and Professor Walther Gotz, The Civilization of the Renaissance, New York: Harper and Row, 1929, 2 vol.

3. André Chastel, Renaissance méridionale, Paris: Librairie Gallimard, 1965.

4. André Chastel, Le grand atelier d'Italie, Paris: Librairie Gallimard, 1965.

5. Enrico Castelnuovo e Carlo Ginzburg, "Centro e periferia," Storia dell'arte italiana, Turin: Giulio Einaudi, 1979, pp. 285-352. Cf. Ferdinando Bologna, La coscienza storica dell'arte d'Italia, Turin: UTET, 1982.

6. Alessandro Parronchi, Studi su la dolce prospettiva, Milan: Aldo Martello, 1964, pp. 583 ff.

7. For an analysis of this treatise see the author's work on Leonardo, as in I.1, note 14, pp. 57-60, 68-86.

8. Christiane L. Joost-Gaugier, "The tuscanization of Jacopo Bellini: Part 1: the relation of Jacopo to problems of the 1420's," Acta historiae artium, Budapest, vol. 23, fasc. 1, 1977, pp. 95-112 and "The tuscanization of Jacopo Bellini: Part II: the relation of Jacopo to problems of the 1430's," Acta historiae artium, Budapest, vol. 23, fasc. 2, 1977, pp. 292-312.

9. André Corboz, Canaletto: una Venezia immaginaria, Milan: Alfieri Electa, 1985, 2 vol. is a penetrating and fundamental study.

10. G. Vasari, Lives, as in note 1 above, vol. II, pp. 44.

11. Ibid., p. II, 141.

12. This sermon is sufficiently remarkable to deserve citation in full. Below I give a translation followed by the original:

May the spirit of divine grace illuminate our senses and our heart. Amen. Reverend lords, venerable fathers, excellent doctors, magnificent men, most keen students of this faculty and other most outstanding citizens. Of all arduous and difficult things, the most difficult is proportion. For it is this alone which penetrates deeply and most high and individual aspects of the trinity and is studied most carefully by the sacred theologues. For this is what is often called "relatio" in their volumes, or "respectus" or even "habitudo." Or in an intellectual discourse it is called by another name, "comparatio." Its acquaintance is something of which divine philosophers are most desirous when preparing metaphysical works for publication. It is ardently followed by natural philosophers such as Socrates, Plato, Aristotle and all the rest when they are treating the nature of things of the universe. Nor is it different with the universe of things above: for surely their debt is to be sought in proportion or "habitudo" just as with inferior things. And if they apply themselves to sacred scriptures, how can they explain the procession of the holy spirit from the father to the son caused by their reciprocal love and are able to express language with the pen, unless they first perceive amongst themselves the relation of the father to the son. This the greatest maker always has in front of his eyes in the disposition of things of the heavens and earth while he disposes the movement and path of the stars and all the planets in most orderly fashion. And this when he established the air above and appended the fundament of the earth and liberated the sources of water and gave to the sea its boundary, placing a law on the waters that they would not overflow their bounds, by which the waters were all together. And what delight would there be for the human race if from so great a diversity of things no proportion arose? As is often said it is variety which delights. Moreover how could one be grouped by the love of the invisible unless one perceived some relation of created to creator? And although no proportion is predicated from the finite to the infinite, it is not denied by the learned that a proportion between them was attained by the sacred. Moreover, the natural philosophers, as mentioned above, sought for the proportions of natural things assiduously, as is found in their texts and above all Aristotle, whose works are at hand more than others. For in his physics the proportion of movements of sound are most subtly examined. And from the ten commandments from which comes the number ten, all physicists are content to deduce one of relation or add spatially something of this sublime investigation of proportion. I shall omit some of the other almost innumerable places where there is most frequent discussion of proportions and proportionalities. Among these it is especially to doctors (as I conclude from De naturalibus) who are most illustrious (and in whose hands is the health of all) that they are noted and must be out of necessity. For hot and cold, dry and wet will not have their proper ratios in their medicine, unless (doctors) have found one of these scales which we have mentioned which, after having proportioned from many qualities, they make one which exhibits characteristics appropriate for the sick body. How would the astronomers who abandoned proportion act other than mindless blind men? Those who feel, speak, as the Egyptians say, such as Ptolemy, Albategni, Alfraganus, Geber, Albumaser and the others who by utmost diligence have reached the forefront of proportion. Similarly the chorographers and cosmographers. And Marinus, whom Ptolemy often attacks, Strabo and the others who have most accurately handed down the positions of the entire orb on our maps could they have put so much into a single book if they had not observed proportion? I ask them to speak of all the architects and inventors of different machines past and present: Pythius, who first designed the temple of Venus in a noble style, Dinocrates, Archimedes, Vitruvius, Frontinus, Vegetius and many others who excelled in the construction of buildings. Do the profuse ruins by which they are remembered bear witness by what means it was they built such things? It is surely that they had proportion as a guide as all their work shows. Why should the most famous painters Apelles, Myron, Polyclytus and others whom the historians name worthy of praise for their perspective in the sight of posterity, if in their figures, their lines and distances did not use proper heights and lengths proportionately. The same applies to stone carvers or sculptors of stone, Phidias, Praxiteles and Appollonius, Nestor and others in such occupations as the aforementioned if they did not use the same proportion most diligently in their marble and statues, just as the missing bits thereof can be easily be put back into place if these [proportions] be found. Likewise, the musicians seek nothing else in their melodies except a proper mode of voice and sounds, that is, sesquialtera, sesquitertia, diapente, diapason and with other proportions of this kind (as Boethius attests) which are in proportion, such that they resonate more sweetly and smoothly in the ears of the audience and bring to them the greatest delight which could scarcely be achieved without proportion and proportionality. Imitating this manner the poets compose their songs (with practically the same measures) the ductyl, iambus, trochee, anapest, tritrachus and procleumaticus, and using the other feet in proportion in their place. And like these, the rhetoricians also assign to their orations the necessary parts and congruent numbers. Grammar, the origin and fundament of all the liberal arts, is also found to observe this when it deals in the teaching of beginners in properly speaking and writing, to end with grave, acute and circumflex accents. Which is also the way of the most just Justinian laws. And [how could] his canons form proper judgments if these did not support both types of justice namely the commutative and distributive? Of which the one, namely, the distributive, belonging to geometry, has been shown to direct so much proportion (as Aristotle in the Ethics and Plato in the Laws and the Republic testify), added to which the just man himself, the judge of the living and the dead will one day give retribution to the human race bringing into proportion to one another the merits and dismerits of all as can clearly be gleaned from the sacred scriptures. The foremost patrons of public affairs of this age also observe the commutative part assiduously when selling things for money or settling or dealing in some other way. And the industry of each of the other mechanical arts has its due proportions which moderate it as experience testifies. But since we are history such, what shall we say of our arithmeticians and geometers who are always usually first among equals, such as Pythagoras and Nicomachus who are cited as having been the inventors of the first numbers among the Greeks, although Boethius and Apuleius were held to have been so among the Romans? Did not these serve some proportions more diligently which (according to Euclid) they called rational? Geometers truly are indifferent in their care for either one, namely, rational or irrational. Finally this proportion is an infinite thesaurus for men by the use of which they are made participants in the friendship of Christ through the gift of discipline of the Commandment. I have disseminated this purpose without deception and to those desirous of it without envy I communicate it showing its virtue most openly. Hence Euclid entered upon the necessary observance of these proportions and proportionalities in order that all that which he said would have greater fruit. Of these same matters he deals most thoroughly in the fifth book, writing thirty four definitions with their premises and then as is his manner their conclusions (which makes up the whole of this book). And he concludes against his adversaries most firmly and irrefutably.

For which reason if someone aspires to any speculation in any faculty of art or science he will approach this fountain whence the rivers of the waters of life flow always. And its ingenuity is to be extolled above the stars. But the case demands that we now come to the text, which begins as follows: A part is

The original is in: Euclid, Euclidis megarensis opera, ed. Luca Pacioli, Venice: Paganinus de Paganinis ,1509, fol. 30:

Sermo habitus per Reverendum patrem M Lucam Paciolum de burgo Sancti Sepulchri Ordo. minorum In ecclesia Sancti Bartholomei. Venetiis. 1508. Die. xi. augusti in quintum Euclidis. Spiritus sancti gratia illuminet sensus et corda nostra. Amen. Arduarum difficiliumque rerum omnium. Reveredi domini uenerandi patres: excellentissimi Doctores: Magnifici viri: Acutissimi cuiuscunque facultatis studentes vosque caeteri prestantissimi ciues : difficillima est proportio. Haec est illa quae sola intima altissimae idividuaeque trinitatis penetrat; et a sacris theologis solertissime investigatur. Haec enim est quae saepius in eorum uolu minibus relatio dicitur: aliquando respectus: nonnunque habitudo. Interdum intellectualis discursus: et nomine alio comparatio nucupatur. Huius notitiam divini philosophi fummopere cupierunt: dum Metaphysicen opera in lucem prodere curarent. Hanc pro viribus naturales prosequti funt: ut Socrates: Plato: Aristoteles: caeterique omnes. Cum de rerum uniuvuersique natura agerent. Non enim aliud in rebus universis superioribus: scilicet et inferioribus quae debita earum adinvicem proportio seu habitudo queritur. Nunquam enim sacris litteris incumbentes: processionem sancti spiritus a patre et filio ex eorum reciproco amore causa tam in lingua calamoque explicare potuissent: nisi prius relationem inter eos patris ad silium: et econtra percepissent. Hanc preoculis summus opifex in caelestium terrestriumque rerum dispositione semper habuit. Dum orbium motus cursusque syderum et planetarum omnium ordinatissime disponeret. Haec quando aethera firmabat sursum: et appendebat fundamenta terrae: et librabat fontes aquarum: et mari terminum suum circundabat: legemque ponens aquis: ne transirent fines suos: cum eo erat cuncta componens. Que nam esset humano generi delectatio si ex tanta rerum diversitate proportio non oriretur? Cum saepe dicatur: varietas est que delectat. Quo pacto insuper in invisibilium raperetur amorem nisi habitudinem quandam creaturae ad creatorem cerneret. Et quamvis finiti ad infinitum proportio nulla esse predicetur: attingentie tamen inter ea proportio a sacris non negatur doctoribus. Naturales autem et ipsi (ut paulo ante diximus) persedulo rerum naturalium proportiones quaesivere: prout in eorum codicibus passim habetur. Presertim Aristotelis cuius opera pre aliis assidue premanibus habentur. Nam inde physico auditu proportionem motuum inter se subtilissime perscrutatur. Et ex decem predicamentis quo numero denario omnes philosophi contenti extitere: unum relationis seu ad aliquod huic tam sublimi indagatrici proportioni seu spatialiter addicauit. Omitto loca alia pene innumerabilia ubi de proportionibus et proportionalitatibus saepissime disseritur. Que omnia ( ut de naturalibus concludam:) medicis praesertim peritissimis (quibus omnium cura commissa est) nota sunt et essere de necessitate debent. Non n calidi et frigidi: humidi et sicci in medelis disponendis rectam rationem habebunt nisi gradum cuiuslibet predictarum nouerint quem postea ex multis proportionando qualitatibus: unam efficiunt egrotam ti corpori debite exhibemdam. Quo et Astronomi proportione relicta agerent: nnone velut amentes ceci que discurrerent. Narrent hii qui sentiunt: dicant egyptii ut Ptolomeus: Ali. Albategni,. Alfagranus, Geber. Albumaseri et ceteri omnes qui proportione previa peritissimi evassere. Qualiter corograpoi cosmographique. Marinus quem saepe Ptolom eus inpugnat Strabo et alii qui totius orbis situm nobis tabulis quibusdam accuratissime tradiderent tot si tanta simul unico libello complecti potuissent? nisi matrem divinum observassent proportionem. Dicant queso architecti omnes et diversarum machinarum inventores prisci et presentes: Pythius qui primus aedem minerve nobiliter architectatus est. Dinocrates: Archimedes: Vitruvius: Frontinus: Vegetius: et alii quamplures quae in aedificiorumque structuris summe excelluerunt: quoque memoriam persuse ruinae adhuc nobis asserunt quo medio talia ederint? Certe proportione duce se omnia prefissecrespondebunt. Quomodo pictores celeberrimi. Appeles, Miron, Policletus et caeteros quos historiae nominant aliquid laude dignum prospectivo aspectu suis posteris reliquissent. Si in eorum figuris liniamenta distantiasque debitas altitudines et latitudines proportionaliter non servassent. Lapicidae quoque seu lapidum sculptores Phidias: Praxiteles. Appollonius. Nestor et reliqui industria tali prediti: non ne eandem diligentissime proportionem marmoreis aeneisque statuis accomodarunt. prout indies frustis talium hinc inde repertis facile datur intelligi. Pariter et Musici :nil aliud in eorum melodiis: armoniisque querunt nisi modum debitum vocum et sonorum: hoc esti: Sesquialtera sesquitertia. Diapente: Diapason: et aliis huiusmodi proportionibus (teste Boetio) proportionatum. ut in auditorum auribus dulcius ac suavius resonent. et summam illis delectationem ingerant: quae sine proportione et proportionalitate minime Causari potest. Quem morem imitando poetae Carmina sua (eisdem fere mediis) Datilo: Iambo: Spondeo: Trocheo: Anapesto: Tribraco. Proceleumatico. Ceterisque proportionis loco utendo pedibus. Componunt. Necnon et rethores (ad istorum instar) Orationum suarum partes debitis. ac congruis numeris assignant. Hoc idem origo et fundamentum omnium liberalium artium grammatica observare videtur: dum normam recteloquendi recteque scribendi discere incipientibus tradit. gravi: acuto: circumflexoque: acentibus terminatam. Qua et via aequissimae sanctiones. Justiniana scilicet et canonica suas recte formarent sententias: si iustitiam utranque commutativam scilicet et destributivam non supponerent. Quarum altera videlicet distributiva penes geometricam tantum proportionem attendi comprobatur (ut in ethicis Aristoteles: et plato inde legibus et republica testantur) iuxta quam iustus iudex vivorum et mortuorum olim humano generi retribuet merita ac demerita omnium adinvicem proportionando ut ex sacris aperte elicitur litteris. Hanc assidue et commutativam observant rerum publicarum fautores dignissimi huius seculi negociatores res pecunia vendendo. emendoque seu quovis alio modo pertractando. Aliarum quoque unaquaeque mecanicarum industria suas debitas habet proportiones ipsam moderantes: experientia teste. Sed dum talia percurimus quid de arithmeticis geometrisque nostris dicemus: qui precipui inter alios semper habitae sunt: ut Pitagoras et Nicomacus: qui prirni numerorum apud graecos inventores fuisse perhibentur: quis apud latinos Boetius et Apuleius habeantur. Non ne hi ceteris diligentius proportionem servant: quam (teste Euclide) rationalem vocant. Geometre vero utrique indifferenter rationali scilicet et irrationali curam adhibent. Hec denique proportio infinitus thesaurus est hominibus quo qui usi sunt participes facti sunt amicitie dei propter disciplinae dona Commendati. Hanc ego proposse sine fictione didici et cupientibus sine invidia communico virtutem eius apertissime ostendendo. harum igitur proportionum et proportionalita tum Euclides necessariam cemens observantiam ut omnium que dixerit fructus uberior habeatur. De his ipsis disertissime hoc in quinto egit. Diffinitiones earundem premittens ac deinceps more suo conclusiones trigintaquator numero. (quibus iste totus complectitur liber) exarando. Et contra adversarium eas firmissime atque inrefragabiliter concludit. Qua propter siquis ad speculationem aliquam quacunque in facultate scientia: arteque: aspirat ad hunc properet fontem: a quo aquae vive semper flumina fluunt. Et super astra eius extolletur ingenium. Sed ut iam ad litteram veniamus res expostulat. Que sic incipit videlicet. pars est.

13. Vasari, Lives, as in note 1 above, p.252.

14. Ibid., p.264.

15. Ibid., p.272.

16. Ibid., vol.II, pp.14-16

17. Ibid., vol. I, p.302

18. Paul Lawrence Rose, The Italian Renaissance of Mathematics, Geneva: Librairie Droz, 1975.

19. Margaret Daly Davis, Piero della Francesca's Mathematical Treatises, Ravenna: Longo editore, 1977.

20. Cf. Chastel, as in note 3, pp. 251-263.

21. See: Henry de Geymüller. Les projets primitifs pour la basilique de Saint Pierre, Paris: J. Baudry, 1875, 2 vol.

22. Wolfgang Lotz, "Das Raumbild in der italienischen Architekturzeichung der Renaissance," Mitteilungen des Kunsthistorischen Instituts in Florenz, Florence, Bd. 7, 1953-1956, pp. 193-226. Cf. also his: Studies in Italian Renaissance architecture, Cambridge Mass., MIT Press, 1977.

23. G. Vasari, Lives, as in note 1 above, Vol. I, p. 326.

24. Giovanni Paolo Lomazzo, 1590, as in I.1, note 35, pp. 16-17, 52, 149-150.

25. See the author's study of Leonardo, as in I.1 note 14 for documentation and analysis of these claims.

26. See Sergio Marinelli, "The Author of the Codex Huygens," Journal of the Warburg and Courtauld Institutes, London, vol. 44, 1981, pp. 214-220, pl. 30-35.

27. Erwin Panofsky, "Die Perspektive als symbolische Form," Vorträge der Bibliothek Warburg 1924-1925. Leipzig, 1927, Taf.XIII, Abb.22.

28. Liliane Brion Guerry, Jean Pélerin Viator. Sa place dans l'histoire de la perspective, Paris: Société d'édition les belles lettres, 1962. (Les classiques de l'humanisme, Vol. VIII).

29. René Descartes et Beeckman, "Correspondence" (Breda, 26 mars 1619), (Copie Ms. Middlebourg, Provinciaal Bibliotheek Zeeland, Journal de Beeckman, fol. 288v) in: Oeuvres de Descartes, ed. Charles Adam et Paul Tannery, Paris: Leopold Cerf, vol. 8, 1908, pp. 156-157:

Et certe, ut tibi nude aperiam quid moliar, non Lullij Artem brevem, sed scientiam penitus novam tradere cupio, qua generaliter solvi possint quaestiones omnes quae in quolibet genere quantitatis, tam continuae quam discretae, possunt proponi.

30. Ibid., p. 157:

alia solvi non posse, nisi cum aliis lineis curvis, sed quae ex unico motu oriuntur, ideoque per novos circinos duci possunt, quos non minus certos existimo et geometricos, quam communis quo ducuntur circuli.

 

31. Rene Descartes, "Trait‚ de l'homme" in: Oeuvres de Descartes, ed. Charles Adam et Paul Tannery, Paris: Leopold Cerf, 1909, vol. XI, p. 130:

Ainsi que vous pouvez avoir vue, dans les grottes et les fontaines qui sont aux jardins de nos Roys que la seule force dont l'eau se meut en sortant de sa source est suffisante pour y mouvoir diverses machines et mesme pour les y faire iouer de quelques instrumens, ou prononcer quelques paroles, selon da diverse disposition des tuyaux qui la conduisent.

32. Jurgis Bultrusaitis, as in I.1, note 10, p. 41.

33. Jean Francois Nicéron, La perspective curieuse, Paris, 1652 (as cited in Baltrusaitis, 1956, p. 41):

Philon le Juif aux livre De specialibus legibus dit expressement en ces termes...que la vraye magie, ou la perfection des sciences consiste en la perspective, qui nous fait cognoistre et discerner plus parfaictement les plus beaux ouvrages de la nature et de l'art et qui a este estimée de tous temps non seulement du commun des peuples, mais encore des plus puissans monarques de la terre.

34. See, for instance, Otto Mayr and Klaus Maurice, The Clockwork Universe: German Clocks and Automata, 1550-1650, New York: N. Watson, 1980. Originally: Die Welt als Uhr, Munich: Deutscher Kunstverlag, 1980; Cf. Klaus Maurice, Die deutsche Räderuhr: Zur Kunst und Technik des mechanischen Zeitmessers im deutschen Sprachraum, Munich: Beck,1976, 2 vol.

35. James Mitchell Collier, Linear Perspective in Flemish Painting and the Art of Petrus Christus and Dirk Bouts, Ph.D., University of Michigan, 1975.

36. For a recent discussion of this problem see Marisa Dalai Emiliani, "Per amore dell'arte della segreta prospettiva..." in: Carlo Pedretti ed., Leonardo il Codice Hammer e la mappa di Imola, Florence: Giunti, 1985, pp. 185-186.

37. Hieronymus Rodler, ed., Perspectiva. Eyn schön nützlich Buchlein...so sich der Kunst des Messens (Perspectiva zu latein genannt)...Simmeren: Hofdruckerei 1531.

38. See: Pierre de la Ramée, (Ramus), Praelectiones in Ciceronis orationis octo consulares una cum ipsius vita per Joannem Thomam Freigium, Basel: P. Pernam, 1580, pp. 338-339.

39. Heinrich Kreisel, Die Kunst des deutschen Möbels, Munich: C.H. Beck, 1968.

40. See Maximilian Bobinger, Alt-Augsburger Kompassmacher, Augsburg: Hans Rösler Verlag, 1966. (Abhandlungen zur Geschichte der Stadt Augsburg. Schriftenreihe des Stadtarchivs Augsburg, 16).

41. Maximilian Bobinger, Christoph Schissler der Ältere und der Jüngere, Augsburg, Basel: Verlag der Brigg, 1954.

Chapter 3

1. This was technically the first discussion of ground-plan and elevation in connection with perspective, although it is frequently assumed that Brunelleschi combined a ground plan and elevation in arriving at his construction. Indeed, Vasari, Lives of the artists, as in I.2, note 1, claims this in vol. 1, p. 272:

He paid great attention to perspective, which was badly understood at the time, many errors being perpetrated, and spent much time over it, but at length he discovered unaided a method of getting it perfectly true; this was to trace it with the ground plan and elevation by means of intersecting lines, a useful addition to the art of design.

Elsewhere, however, Vasari suggests that Paolo Uccello was the innovator, in his Lives, vol. 1, p. 232:

But Paolo, without ever wasting a moment was always attracted by the most difficult things of art, and brought to perfection the method of representing buildings, to the tops of their cornices and roofs, in perspective from their plans and elevations. This was done by intersecting lines, diminishing at the centre; the point of view, whether high or low, being first decided. He laboured so hard over these difficulties that he invented a method and rules for planting figures firmly on their feet and for their gradual foreshortening and diminution in proportion as they recede, a matter that was previously left to chance. He also discovered the method of tracing the ribs and arches of vaulting, the foreshortening of floors by diminishing the receding beams, and the way to make round columns follow the turn made by the sharp corner of a house, doing this from a ground plan.

2. For an analysis, see the author's work: Leonardo, Studies I, as in I.1, note 14, pp.57-60, 68-86.

3. See: Gezenius ten Doesschate, De derde commentaar van Lorenzo Ghiberti in verband met de middeleuwesche optiek, Proefschrift, Utrecht, 1940.

4. For instance, Alberta, On Painting, as in I.1, note 13, pp. 47 and 49:

Nor is this the place to discuss whether vision, as it is called resides at the juncture of the inner nerve or whether images are formed on the surface of the eye as on a living mirror...

Let us omit the debate of philosophers where the original source of colours is investigated.

Both of these passages are in the Latin text and are omitted altogether from the Italian version.

5. This quote is from the first English edition: Sebastian Serly, The second book of architecture, London: Printed for Robert Peake, 1611, fol. Ai v.

6. Pietro Accolti, Lo inganno de gl'occhi, Florence: Appresso Pietro Cecconcelli 1625, p. 116:

E cosi grande l'autorita di Vitellione, unico, e principal capo della scuola de perspetttivi, che chiunque ardisca pronunciare egli haver falsamente, o dimostrato, o insegnato, puo di facile esser reputato, temerario, o ardito o molto...

7. Ibid., p. 116: "Falsa dimostrazione, e meno vera dottrina di Vitellione circa l'obliquo passaggio de i lumi, Cap. XVII."

8. Ibid., p. 13: "Che la prospettiva non sia altro in effetto che una rappresentativa sezione della piramide visiva."

9. Giacomo Barozza, il Vignola, Le due regole della prospettiva pratica, ed. Egnazio Danti, Rome, 1583, preface:

abbiamo oggi tre libri scritti a mano, eccellentissimamente disegnati.

10. Ibid., p. 49:

Perche oltre alla descrittione delle figure rettilinee, apporta gran commodita al prospettivo il saperle trasmatare d'una nell'altra, ho voluto in queste tre seguenti propositioni mostrare il modo secondo la via commune non solamente di trasmutare il circolo e qual si voglia figura rettilinea in un altra, ma anco di accrescerle e diminuirle in qual si voglia certa proportione, accio in questo libro il prospettivo habbia tutto quello, che a cosi nobil pratica fa mestiere.

11. Guidobaldo del Monte, Perspectivae libri sex, Pesaro: Apud Hieronymum Concordiam, p. 1:

Architecturam, atque picturam reliquas omnes anteire artes, quae citra manuum usum sola ingeniorum applicatione, atque solertim, quod intendunt, moliri, ac perficere nequeunt (quae propterea mechanicae appellantur) nemini certe egregia earum opera consideranti, ambigendum censeo.

12. Simon Stevin, Derde stuck der wisconstighe ghedachtnissen van de deursichtighe, Leiden: Ian Bouwensz, 1605:

maer willen een voorghestelde verschaeulicke saeck volcomelick afteyckenen, met kennis der uirsaken en sign wisconstich bewijs.

13. The quote has been adapted from the first English edition of Serlio, as in note 5, fol. Aiv.

14. Pietro Cataneo, I quattro primi libri, Venice: Aldus, 1554, fol 1r:

Quel che piu facci di bisogni all architetto e di quanto importanza gli sia l'essere buono prospettivo.

15. Ibid:

Ma se l'architetto non sera prospettivo, non potra mai cosi bene ne

honorarsi, ne mostrare per disegno il suo concetto, per eccellente disegnatore chei si fusse: e da se stesso sconscera di questa importanza gli sia il non essere nella prospettiva ben prattico.

16. The phrase il suo alzato per ordine di prospettivo appears in the following chapter headings I.VIII, IX, X, XIII, XVII, XVIII, XIX; III.introduction, II, VII, IX. Cf. Ibid, 44v:

La figure qui appresso che segue rappresenta la meta del tempio nella parte interiore di ordine Corinto, tirato dalla sua pianta passata per ordine di prospettiva.

17. Vincenzo Scamozzi, L'Idea della architettura, Venice: Expensis auctoris, 1617, 47:

Poi la prospettiva, serve per rappresentare per via di linee artificiali tutte le cose, come dice anco Vitruvio, stando in certo determinato luogo: corrispondono a raggi nostri del vedere naturale, in modo tale che apportano a gli occhi nostri de specie e l'imagine vere, de gli edifici, che sono disegnati in iscorcio nelle scene e altrove.

18. Ibid.:

e certo e mirabil cosa il vedere, che il piano delle tele, o delle tavole, con colori, siano talmente ben disposte, e lineate dall'arte, che a quelli, che le mirano paiano che siano di rilievo, e piu alto, e piu basso.

19. Ibid.:

e di questa facolta sino nella nostra prima gioventu ne abbiamo scritto sei libri, ne quali e molto numero di disegni, e cosi delle cose superficiali e in piano, come de corpi e parti di cinque ordini, i quadri speriamo in Dio di metterli in luce, dopo questa nostra lunga e faticosa opera d'architettura.

20. Cf. Rodler (1531). An example at the turn of the 17th century is Jan Vredeman de Vries, Perspective, id est, celeberrima ars inspicientis...perutilis ac necessaria, omnibus pictoribus, sculptoribus, statuariis, fabri ferrariis, architectis, inventoribus caementariis, scrinariis, fabrilignariis, et omnibus artium amatoribus, qui huic arti operam dare volent, majori cum voluptate, et minori cum labore, Leiden: Hondius, 1604.

21. Jean Dubreuil, La perspective pratique, necéssaire a tous peintres, graveurs, architectes, brodeurs, sculpteurs, orferres, tapisseurs et autres qui se meslent de desseigner, Paris: Chez Jean Du Puis, 1643-1649. 3 vol.

22. Giovanni Battista Benedetti, Diversarum speculationeum liber, Turin: Apud haeredem Nicolai Bevilaquae, 1585, p. 119: "hunc solum esse verum modum."

23. For a more detailed discussion of these developments see the appendix to the author's work on Leonardo, as in I.1, note 14.

24. Plato, Timaeus, trans. H.D.P. Lee, Harmondsworth: Penguin, 1965, pp. 72-78.

25. See: Regiomontanus, "Commensurator," ed. Wilhelm Blaschke und Gunther Schoppe, in: Abhandlungen der Akademie der Wissenschaften und der Literatur in Mainz, Mathematisch-naturwissenschaftliche Klasse, Wiesbaden, Nr. 7, Jahrgang, 1956.

26. Wenzel Jamnitzer, Perspective corporum regularium...durch einen newen behenden und gerechten Weg der vor nie in gebrauch ist gesehen, Nürnberg, 1567.

27. Ibid.: wie auss denselbigen funff Corpern one Endt gar viel andere Corper mancherley Art und gestalt gemacht unnd gefunden werden mügen.

28. Julius Hoffman, Die Kupferstiche des Meisters PP mit der Schlinge, Vienna: Gesellschaft fur vervielfältigende Kunst, 1911.

29. See Linda Dalrymple Henderson, The Fourth Dmension and non-Euclidean Geometry, Princeton: Princeton University Press, 1983.

30. This problem has been studied in an unpublished essay by Geoffrey Smedley, "Notes concerning Piero della Francesca's Prospectiva pingendi," unpublished Typescript, Vancouver: University of British Columbia, c. 1986.

31. Giovanni Paolo Lomazzo, as in I.1 note 36, 1585, pp. 270, 175, cf. 100, 320; 1590 as in I.1, note 35, 108, 150, cf. 16.

32. See the author's study of Leonardo, as in I.1, note 14, pp. 202-239

33. See: Sergio Marinelli, as in I.2, note 26.

34. E.g. Serlio, as in note 5, Bk. III, A1v-A2r:

with part of the sciographies of the most famous antiquityes of Rome...and that I may goe forward orderly in these antiquities. The first figure shall be the ichnography. The second, the orthography, the third, the sciography.

This led Barbaro, 1568, p. 129-130 to note:

che la prospettiva era molto necessaria all'architetto, e cosi hanno interpretato quella parola sciographia per la prospettiva, la quale e come una adombratione.

Molti anche hanno letto scenographia, in luogo di sciographia, & hanne inteso lo istesso, cioe la descrittione delle scene, laquale ricerca miribilmente l'uso della perspettiva...

These debates continued in the seventeenth century. See, for instance, Vitruvius, De architectura, Amsterdam: L. Elzevirium, 1649, page 6, note b or Claude Perrault's edition of Vitruvius, Les dix livres d'architecture, Paris, 1684, p. 10, note 7 beginning:

Barbaro a mis sciographie au lieu de scenographie que Hermolaus Barbaro en ses gloses sur Pline a restitue avec beaucoup de raison, puisque la definition que Vitruve apporte du mot dont il s'agit, et qui est proprement celle de la perspective convient tout a fait au mot de la scenographie...

Professor Werner Oechslin (Zürich) is working on this problem.

35. Daniele Barbaro, La pratica della perspettiva, Venice, 1568, p. 129-158:

parte quarta, nella quale si tratta della scenographia, cioe descrittione delle scene.

36. Giacomo Barozzi, il Vignola, Le due regole, ed. Egnazio Danti, Rome: Zanetti, 1563, pp.

37. Antonio Manetti, The Life of Brunelleschi, ed. Howard Saalman, University Park: Pennsylvania State University Press, 1970, p. 54.

38. Ibid., p. 54:

Since they undertook excavations to find the junctures of the membering and to uncover objects and buildings in many places where there was some indication, they had to hire porters and other laborers at no small expense. No one else attempted such work or understood why they did it.

On the question of ruins see: Paolo Arrigoni e Achille Bertarelli, Piante e vedute di Roma e del Lazio, Milan: Castello Sforzesco, 1939.

39. Daniele Barbaro, as in note 35, p. 159:

Parte quinta, nella quale si espone una bella e secreta parte di perspettiva.

40. Egnazio Danti, as in note 36, p.

41. See Carlo Pedretti, "Un ritratto anamorfico di Francesco I, di probabile invenzione Vinciana," in: Ibid, Documenti e memorie rigunrdanti Leonardo da Vinci a Bologna e in Emilia, Bologna: Editoriale Fiammenghi, 1953, pp.121-123.

42. Julius Deininger, Eine neue Theorie der malerischen Perspektive und deren praktischen Resultate, Vienna: Gerlach und Weidling, 1914, p.14:

Daraus aber folgt, dass es einzig und allein auf einer solchen Kugeloberfläche möglich ist, alle diese Längenmasse, das heisst alle perspektivischen Dimensionen in ihren richtigen Zusammenhänge und ihrer richtigen Grösse graphisch darzustellen.

 

43. Adelbert Ames, Jr. and C.A. Proctor, "Dioptrics of the Eye," Journal of the Optical Association of America, Rochester, vol. 5, 1921, p. 22. The article covers pp. 22-84.

44. Ivan Jobin, Ligne droite on ligne courbe? Cone ou sphère optique, Montreal: Editions Albert Levesque, 1932, p. 13:

La seconde partie s'attache à démontrer que la ligne courbe, déterminé par les principes de la sphère optique, constitue, aujourd'hui une théorie de vision supérieure à celle de la ligne droite établie selons les principes du cône optique.

45. For an introduction to this history see Massimo Scolari, "L'idea de modello," Eidos, Asolo, vol. 2, 1989, 16-39.

46. Cf. Peter Jeffrey Booker, A History of Engineering Drawing, London: Chatto and Windus, 1963.

47. Ptolemy, L'optique de Claude Ptolémée, ed. A. Lejeune, Louvain: Publications universitaires de Louvain, 1956, p. 74 (II. 124):

Et ideo pictores domorum constituunt colores rerum quas remotas volunt ostendere, aeros latentes.

Lejeune notes (104): "C'est le principe de la perspective aérienne. Meme sujet dans un intérressant fragment papyrologique publie par K. Wessely, dans Wiener Studien, Vienna, XIII, 1891, pp. 319-323."

48. Pompilius Azalus, De omnibus nebus naturalibus quae continentur in mundo, Venice: Apud Octavianum Scotum, 1544, fol. 74v:

Ab hae naturali experientia, ars pictoria optimus canones accepit, ut in libello and Jacopum Bellinum venetum pictorem insignem certi descripsi, quibusque modis colores obscuros et claros opponere sciret, tali cum ratione quod non solum unius imaginis partes relevatae videruntur in plano depictae, verum extra manum vel pedum porrigere crederentur inspectae, et eorum quae in eadem superficie hominum animalium vel montium equantur quaedam per miliaria distare apparent atque ejusmodi. Is quidem pingendi docet propinqua claris, remota obscuris, mediasque permiutis sub coloribus tingi debere.

49. Cf. Janis Clearfield-Bell, Color and Theory in Seicento Art, Zaccolini's Prospettiva del Colore and the Heritage of Leonardo, Ph.D., Brown University, 1983.

50. Louis Clement de Brunel de Varennes, Métroscopographie, Paris: Troyes, 1830, p. 19:

Nous divisons la perspective en deux principales sections. La premiere est la perspective linéaire; la seconde est la perpective aérienne, dans laquelle sont comprises la théorie des ombres, celle des reflets, et la réflexion des objets dans l'eau et sur les surfaces polies.

51. Armand-Denis Vergnaud, Manuel de perspective, Paris: Roret, 1835, pp. 74-75:

Définition - La perspective à pour but de représenter sur une seule et meme surface, l'ensemble et les détails des objets que la nature répartit à distances inégales sur des surfaces variées à l'infini: il est, pour atteindre ce but, deux parties bien distinctes dans la perspective. L'une doit déterminer les contours apparens des corps et leurs positions respectives sur les differentes surfaces ou ils se trouvent, l'autre doit saisir la couleur meme de ces objets, avec toutes les modifications que lui font subir et les accidens de la lumière et les couches plus ou moins épaisses d'air atmosphérique qui les séparent les uns des autres. La première est une science positive, ou l'on est guid‚ rigoureusement par les principes les plus simples de la géométrie: c'est la perspective linéaire, sans laquelle on ne peut être dessinateur. La seconde, qui semblerait au premier abord pouvoir s'obtenir d'une manière non moins rigoureuse, à l'aide de la géométrie et de la physique, constitue la perspective aérienne, sans laquelle on ne peut être peintre: mais il ne faut pas espérer en ateindre la magie des couleurs et la sublimité, si l'on n'est pas doué de cette sensibilité exquise et de cette chaleur d'imagination, étincellés du feu sacre sans lequel il ne peut exister de véritable artiste. En un mot on peut, à l'aide de la perspective linéaire, produire de belles statues dont les formes seront pures et agréables, mais pour lesquelles la perspective aérienne ne cessera jamais d'être le souffle créateur de Prométhée.

52. Barnabé Brisson, Théorie des ombres et de la perspective, Paris: Bachelier, 1838, p. 24:

On voit qu'ici, comme dans la théorie des ombres, on doit admettre deux parties distinctes: l'une est purement géométrique, et son objet est de déterminer d'une manière précise sur le tableau la position de chaque point représenté; l'autre à pour objet la recherche de la teinte d'ombre et de lumière qu'on doit donner à chaque partie du tableau, et c'est par des considérations physiques qu'on peut en général la traiter. Cette dernière partie, qu'on designe sous le nom de Perspective aérienne, rentre entièrement dans le cercle des recherches qui nous essaierons d'exposer plus tard, pour completer la théorie des ombres.

53. Jules De La Gournerie, Traité de perspective linéaire, Paris: Dalmont et Dunod, 1859, p.1:

La Perspective est l'art de représenter les objets sur un tableau en conservant leur apparence. Elle est linéaire ou aérienne, suivant qu'elle s'occupe des formes ou de la coloration. Il ne sera pas question dans cet ouvrage de la perspective aérienne.

54. Ernst Wilhelm Brücke, Principes scientifiques des beaux arts, Paris: Germer, Baillière et Cie., 1878, particularly, pp. 62-65.

55. Dictionnaire de pédagogie et d'instruction primaire, ed. F. Buisson, Paris: Hachette, 1882, vol. 2, p. 1555:

Perspective Pratique - La perspective est l'art de reproduire sur une surface plane l'aspect des objets tels qu'ils se présentent à nous dans l'espace.

La perspective linéaire étudie la réproduction des contours des objets; la perspective aérienne s'occupe plus spécialement des modifications qu'apporte aux ombres et aux teintes la couche d'air interposée entre les objets et l'oeil du spectateur.

56. Louis Delaistre, Cours complet de dessin linéaire, Paris: Gauthier Villars, 1894, p.1:

La perspective, considérée dans son ensemble, est l'art de représenter les objets sur une surface plane suivant leurs effets d'optique; c'est-à-dire selon les lois de la vision et celles de la lumière, ce qui fait qu'on la divise en deux classes: l'une que l'on nomme linéaire, l'autre que l'on nomme aérienne. La perspective linéaire se fait par les lignes seules. La perspective aérienne se fait par la dégradation des couleurs résultant de l'éloignement de la lumière comme de l'éloignement successifs des objets, ainsi que de la qualité plus ou moins intense des vapeurs terrestres qui s'interposent entre l'oeil du spectateur et ces memes objets.

57. Leon Battista Alberti, On painting, as in I.1, note 13, p. 82:

But I should like the [highest level of attainment] in industry and art to rest, as the learned maintain, on knowing how to use black and white. It is well worth all your study and diligence to know how to use these two well because light and shade make things appear in relief.

The original reads:

Tutta la somma industria e arte sta in sapere usare il biancho e il nero, pero che il lume e ombra fanno parere le cose rilevate.

58. Piero della Francesca, De prospectiva pingendi, ed. G. Nicco Fasola, Florence: G.C. Sansoni, 1942, p. 63:

Colorare intendiamo dare i colori commo nelle cose se dimostrano, chiari et scuri secondo che i lumi li devariano.

59. See the authors study of Leonardo, as in I.1, note 14., pp.278-305.

60. Alberti, as in I.1, note 13, pp. 71.

61. For an introduction to these problems see: Thomas Da Costa Kaufmann, "The perspective of shadows: the history of the theory of shadow projection," Journal of the Warburg and Courtauld Institutes, London, Vol. 32, 1975, pp. 258-287.

62. Domenico Tessari, Applicazioni della geometria descrittiva. Vol. 1. La teoria delle ombre e del chiaro scuro, ad uso delle universita...degli ingegneri, architetti e disegnatori, Turin: Camilla e Bertolero, 1880.

63. E.g. J. J. Gibson, The Ecological Approach to Visual Perception, Boston: Houghton Mifflin Co., 1979.

Chapter 4

1. As Panofsky noted in "Die Perspektive als symbolische Form," as in I.2 note 27, p. 291, the term goes back to Boethius, Analytica posteriorum Aristotelis, Interpretatio, I.7, I.10, (Opera, Basel 1570, pp. 527, 538).

2. See, for instance, Aristotle, Analytica posteriora, trans. G.R.G. Mure in: The Works of Aristotle, ed. W.D. Ross, London: Oxford University Press, 1928, (Bk. I.13), 78b 35-79a 5, 10-20, Cf. Ibid., vol. II, Physica, trans. R. P. Hardie and R. K. Gaye, 1930, 194a 9-12.

3. On the question of mediaeval classification of the sciences see James A. Weisheipl, O.P., "The nature, scope and classification of the sciences," Studia mediewistyczne, Warsaw, Vol. 18, 1977, pp. 85-101 and his "Classification of the sciences in mediaeval thought," Mediaeval Studies, Toronto, vol. 27, 1965, pp. 54-90.

4. On the history of the seven liberal arts see: Joseph Mariétan, Problème de la classification des sciences d'Aristote à Saint Thomas, Thèse, Paris, 1901.

5. John Pecham and the Science of Optics. Perspectiva communis, ed. David C. Lindberg, Madison: University of Wisconsin Press, 1970, p. 60:

inter physice considerationis studia lux iocundius afficit meditantes. Inter magnalia mathematicorum certitudo demonstrationis extollit preclarius investigantes. Perspectiva igitur humanis traditionibus recte prefertur, in cuius area linea radiosa demonstrationum nexibus complicatur, in qua tam physice quam mathematice gloria reperitur utriusque floribus adornata.

6. Gregorius Reisch, Margarita philosophica, Basel: Michael Furterius, 1517, fol. aiiv:

1 Grammaticae, 2 Dialecticae, 3 Rhetoricae, 4 Arithmeticae, 5 Musicae,

6 Geometrie elementa, 7 Astronomie, 8 Naturalis philosophie principia,

9 Originem primordialem et productionem omnium verum naturalium. Alchimie, 10 Anime vegitative et sensitive, 11 Anime rationalis, 12 Philosphia moralis.

7. Joachim Fortius Ringelbergius, Opera, Lyons: Apud Gryphium, 1531:

Grammaticen, Dialecticen, Rhetoricca, Mathematicen (Sphaera, Institutiones astronomiae, Cosmographia, Liber de tempore, Tabula de tempore, Optice, Chaos Mathematicum), Divinatio, Communis cuiusdam naturae sunt (Chaos, Experimenta, Liber de homine).

8. John Dee, Mathematicall Praeface annexed to: Euclides, The Elements, trans. H. Billingsley, London: I. Daye, 1570. Cf. John Dee, The Mathematicall Preface to the Elements of Geometrie of Euclid of Megara, ed. A. Debus, New York: Science History Publications, 1975.

9. Ioannis Valentini Andreae, Collectaneorum mathematicorum decades XI. Centum et decem tabulis aeneis exhibitae, Tübingen: Typis Iohan Alexandri Cellii, 1614:

Geometrica, Arithmetica, Statica, Astronomica, Gnomonica, Authomatica, Optica, Architectonica, Munitoria, Mensurata, Lineata.

10. Robert Fludd, Utriusque cosmi maioris scilicet et minoris metaphysica, physica atque technica, Oppenheim: aere Johan-Theodori de Bry, 1617, fol. 3:

Arithmeticam, Musicam, Geometriam, Perspectivam, Artem Pictoriam, Artem Militarem, Motus Scientiam, Temporis Scientiam, Cosmographiam, Astrologiam, Geomantiam.

11. Josephus Blancanus, Sphaera mundi, Bologna: typis S. Bonomij, 1620, pp.388-390:

1 Geometria, 2 Arithmetica, 3 Optica, seu Perspectiva, 4 Mechanica, 5 Musica, 6 Astronomia.

12. Ibid., p.390:

1 Geometria practica, 2 Arithmetica practica, 3 Perspectiva practica,

4 Mechanica practica, 5 Musica practica, 6 Astronomia practica.

13. Johann Ciermanns, Annus positionum mathematicarum quas defendit ac demonstravit Perill. Dom. D. Wolffgangus Philippus Iacobus Unverzagt, Baro de Ebenfurt, Louvain: In Collegio Societatis Iesu, 1641, Ordo disputationum:

Mense October Geometricae

Septembri Chronologicae

Optics, he further subdivided into:

Perspectivae, Orthographica, Scenographica, Practica, Compendiosa, Sciographica, Curiosa, Problemata.

14. Daniel Schwenter, Deliciae physico mathematicae, Nürnberg: Jeremiae Dumlers, 1651:

XVI. Chymia und andere Kunsten

15. R.P. Claudii Francisei Millet Dechales, Cursus seu mundus mathematicus, Lyons: Apud Anissonios, 1690, 4 vol, Tractatus:

I Euclidis libri XIV

XXXI Kalendarium

16. Catalogi librorum impressorum bibliotecae regiae, Göttingen, tomus XI.V, Mathematicae:

1. Geometria speculativa et practica, Arithmetica, Algebra

16. Artes mechanicae et illiberales

17. Pierre Hérigone, Cursus mathematicus...nova, Paris: Piget, 1644:

T.1 Euclides Elementorum lib. 15 appendicem geometriae

T.5 Continens opticam, catoptricam, dioptricam, perspectivam, sphaericorum trigonometriam, theoricas planetarum...gnomonicam et musicam.

T.6 Continens geometricas aequationum cubicarum purarum, atque affectarum effectiones

18. Jacques Ozanam, Cours de mathematique, Paris: Jombert, 1697:

T.1 Introduction aux mathématiques et les elemens d'Euclide

T.5 La géographie et la gnomonique

19. Christian von Wolff, Elementa matheseos universae, Halle: Rengerianum, 1730-1738:

T.1 Que commentationem de methodo mathematica, arithmeticam, geometriam, trigonometriam planam et analysin, tam finitorum quam infinitorum

T.2 Mechanicum cum statica, hydrostaticam, aerometriam atque hydraulicam

T.3 Opticam, perspectivam, catoptricam, dioptricam, sphaerica et trigonometria sphaericam atque astronomiam tam sphaericam quam theoricam

T.4 Geographiam cum hydrographia, chronologiam, gnomonicam, pyrotechniam, architecturam militarum atque civilem.

20. Johann Nikolaus Frobes, Encyclopaediae mathematicae memorialis, Helmstedt: Schnorr, 1743-1746:

Pt. 1 Succincta matheseos purae, hoc est arithmeticae, geometriae ac

trigonometriae delineatio

Pt. 6 Succincta geographiae, chronologiae et gnomonicae...

21. Heinrich Wilhelm Clemm, Mathematisches Lehrbuch, Stuttgart: bey Johann Benedict Mezler, 1764:

Der arithmetischen Wissenschaften

viertes Hauptstuck, von der Perspectiv

22. Johann Nikolaus Martius, Unterricht in der natürlichen Magie, völlig umgearbeitet von Johann Christian Wiegleb, Berlin and Stettin: bei Friedrich Nicolai, 1786, etc.:

I Elektrische Kunststucke

VIII Karten "

23. Abraham Gotthelf Kästner, Anfangsgründe der angewandten Mathematik, Gottingen: Vandenhoek und Ruprecht, 1792:

Abth. 1 Mechanische und optische Wissenschaften

Abth. 2 Astronomie, Geographie, Chronologie und Gnomonik

24. Johan Friedrich Lorenz, Grundriss der reinen und angewandten Mathematik, Helmstedt: Fleckeisen, 1798-1800:

Th. 1 Grundriss der Arithmetik und Geometrie

Th. 3 Grundriss der allgemeinen Grossenrechnung

25. We have made no attempt here to be exhaustive. There was another tradition in the eighteenth century which skipped optics altogether in lists of the mathematical sciences. For our purposes the following five examples will suffice:

Allain Manesson Mallet, La géométrie pratique, Paris: Anisson, 1702:

T.4 La steréométrie

Benjamin Hederich, Anleitung zu den furnehmsten mathematischen Wissenschaften, bennantlich der arithmetica, geometria, architectura, astronomia und gnomonica, Wittenberg: Zimmermann, 1710.

Jakob Hermann, Abrégé des mathématiques, St. Petersburg: Académie impériale des sciences, 1728, 2 vol.:

T.3 La fortification

Abbe Deidier, La mesure des surfaces et des solides par l'arithmétique des infinis et les centres de gravité, Paris: Jombert, 1740-1741:

S.1 Le calcul différentiel et le calcul expliqués et appliqués à la géométrie

S.2 La méchanique générale contenant la statique, l'airométrie,

l'hydrostatique et l'hydraulique

25. Georg Gottlieb Schmidt, Anfangsgründe der Mathematik, Frankfurt: Varrentrapp, 1814-1816:

Abth. 1 Statik, Hydrostatik, Aerostatik und Mechanik fester Körper

Abth. 2 Hydraulik und Maschinenlehre

26. John Dee, Mathematical Praeface, as in note 8 above:

27. Leonard Christoph Sturm, Tractatus de natura et constitutione matheseos, Frankfurt: Schrey, 1706, p. 308.

Generalis Lucis & umbra

Catoptrica

Dioptrica

28. Paul Guldin, De Centro gravitatis, Vienna: G. Gelbhaar, 1635-1641, p. 20:

Proprie Optica, ut Perspectiua Orthographica

Diocatoptrica

29. Theodosius Haesel, Geistliche Perspectiva, Dresden: W. Seyffert, 1652: Brevis synopsis disciplinarum mathematicarum:

mesoptica vel per Aerem

30. Johann Christoph Adelung, Kurzer Begriff menschlicher Fertigkeiten, Leipzig: Christian Gottlieb Hertel, 1781, pp. 258-258:

Die Perspektive theilet sich übrigens in die Linearperspecktive, welche durch Hülfe der Geometrie die richtige Verkürzung der geraden Linien, z.B, an den Theilen eines Gebäudes lehret; in die Luftperspective, welche ganz in das Fach des Malers einschlägt, und Licht und Schatten nach den Veränderungen bestimmen lehret, welche die Farbe der Luft in einer gewissen Entfernung an den Körpern und ihren Farben hervorbringet; und in die Spiegelperspective, welche unordentlich und verzerrt scheinende Figuren zeichnen lehret, welche die sphärischen, komischen und anderen Spiegel wieder in ihrer ordentliches Gestalt darstellen.

I am grateful to Professors Loris and Maria Rita Sturlese for this quote.

31. These are cited in note 13 above.

32. Wiegleb, as in note 22 above, vol. III. Optische Kunststücke:

Instrumente und Maschinen zum Zeichnen.

33. Julius von Schlosser, Die Kunstliteratur, as in Introduction, note 2, 1924, (IV.1), p. 227; cf. 1977, ed. Kurz, pp. 259-260.

34. Erwin Panofsky, "Die Perspektive als symbolische Form," as in I.2, note 27, p. 274.

35. Leonardo da Vinci, Treatise on Painting [Codex Urbinas 1270], trans. A. Philip McMahon, Princeton: Princeton University Press, 1956, vol. II, fol. 231r:

Delle ombrosita et chiarezze de monti...Prospettiva commune.

36. On this copy of John Peckham's Perspectiva communis, now in Milan, Bibliotheca Ambrosiama, Cod. Inc. 1105 see: Zenale e Leonardo. Tradizione e rinovamento della pittura lombarda, ed. Mauro Natale, Alessandra Mottola Molfino e Marisa Dalai Emiliani, Milan: Electa, 1983, p. 165.

37. Louis Leger Vallée, Théorie de L'Oeil, Paris: Baillière, 1844, pp.ix-x:

Tel était l'état de la science lorsque, en 1821, nous publiames le Traité de la science du dessin. Dans cet ouvrage, nous considerons la peinture, le dessin, dans ses différents genres, et g‚n‚ralement l'art d'imiter les objets de manière à produire plus ou moins d'illusion, comme une application des règles au moyen desquelles on peut tromper l'oeil. La théorie de la vision sert donc de base à notre traité ou elle est exposée avec quelque détail, et souvent envisag‚e sous des aspects nouveaux. Dans cet ouvrage, dont la perspective et les ombres sont des parties importantes, la géométrie nous guide constamment, et elle nous a conduit à une explication de l'achromatisme de l'oeil fondée sur la non-homogenéité du corps vitre.

38. John Ruskin, The Elements of Perspective, London: Smith, Elder and Co., 1859, pp. 1-3.

39. Ibid., pp. 99 ff.

40. Armond Théophile Cassagne, Pratique de la perspective, Paris: Claude Fouraut et fils, 1879, pp. 18, 50.

41. Jules De La Gournerie, Traité de perspective linéaire, Paris, Dalmont et Dunod, 1859, pp. xix-xx:

La Perspective est un art graphique spécial; elle présente des difficultés pratiques qui lui sont propres et qui, dans le courant de plusieurs siècles, ont occupé un grand nombre de savants et d'artistes. Les auteurs qui l'ont traitée comme une simple application de la Géométrie descriptive n'ont pas pu lui donner les développements necéssaires. Il y a d'ailleurs lieu de croire que plusieurs d'entre eux avaient dédaigné‚ d'étudier les anciens ouvrages. Les élèves de Monge croyaient, en effect, presque tous, que les arts graphiques ne presentaient avant leur maître qu'incertitude et confusion. On trouve dans les écrits de plusieurs des plus célèbres d'entre eux des assertions tout a fait érronées à cet égard.

42. See, for instance, Hermann von Helmholtz, Handbuch der physiologischen Optik, Hamburg, Leipzig: Teubner, 1856.

43. Panofsky, as in I.2, note 27, pp. 265, 301. Cf. the author's: "Panofsky's perspective: a half century later" in: Marisa Dalai Emiliani, ed., La prospettiva rinascimentale, Florence: Centro Di, 1980, p. 567.

44. A history of professions mentioned in title pages would make a very interesting topic of study.

45. For a list of model books see: R. W. Scheller, A History of Mediaeval Model Books, Haarlem: De Erven F. Bohn N.V., 1963.

46. The standard edition remains: Villard de Honnecourt, Kritische Gesamtausgabe des Bauhüttenbuches Ms. Fr. 19093 der Pariser National Bibliothek, ed. R. Hahnlohser, Vienna: A. Schroll, 1935. 2nd ed. Graz: Akademische Druck und Verlagsanstalt, 1972.

47. For an introduction to this vast literature see: Leonardo Olschki, Geschichte der neusprachlichen wissenschaftlichen Literatur, Heidelberg: Winter, 1919-1922, 2 vol. and Bertrand Gilles, Les ingénieurs de la Renaissance, Paris: Hermann, 1964.

48. For a good introduction see Lawrence Wright, Perspective in Perspective, London: Routledge and Kegan Paul, 1983. The deeper problems involved in these distinctions have greatly interested my mentor Sir Ernst Gombrich: e.g., Art and Illusion, Princeton: Princeton University Press, 1960.

49. For an introduction see Edmond R. Kiely, Surveying Instruments. Their History-Classroom Use. New York: Bureau of Public Teachers College, 1947 (National Council of Teachers of mathematics 19th Yearbook).

50. Franz Reuleaux, Theoretische Kinematik Grundzüge einer Theorie des Machinenwesens. Braunschweig: Vieweg, 1875. English trans. Alex B.W. Kennedy, Kinematics of Machinery. Outlines of a Theory of Machines, London: Macmillan, 1876.

51. Very little has been done on the history of this subject. Useful is Booker, as in I.3 note 46 above and a dissertation by Joachim Kuns, Betrachtungen zur Geschichte der technischen Zeichnungen, Dissertation, Rheinisch-Westfälischen Technischen Hochschule, Aachen, 1980.

52. Pierre Larousse, Grand dictionnaire universel du XIX siecle, Paris: Administration du grand dictionnaire universel, 1870, vol. 6, p. 592:

Dessin lineaire. Le genre se divise en plusieurs branches suivant le but que l'on se propose et la nature des objects a representer. Il comprend le tracé des épures de géometrie élémentaire, descriptive et analytique; la perspective ordinaire et isom‚trique; les dessins d'architecture et de machines la topographie.

53. Leonardo da Vinci, as in note 35, fol. 167-171: "De panni."

54. Jost Amman, Kunstbüchlein, Frankfurt: Romanus Beatus in Verlegung Johann Feyrabends, 1599. A complete collection of the illustrations of Amman is painstakingly being done by Dr. Ilse Franke.

55. E.g., Hieronymus Cock, Compartimentorum quod vocant multiplex genus lepidissimis historiolis poetarumque fabellis ornatum, Antwerp: Cock, 1560. Cf. Iacques Floris, Veelderhande cierlijke compertementen..., Antwerp: Hans Liefrinck, 1564.

56. Heinrich Vogtherr, Kunstbüchlein, Strasburg: Christian Müller, 1572. Reprint: Zwickau: Verlag von F. Ullmann, 1913, (Zwickauer Facsimiledrucke, Nr. 19).

57. H. Soden-Smith. A List of Books and Pamphlets in the National Art Library, South Kensington Museum on Drawing, Geometry and Perspective, London: Eyre and Spottiswoode, 1888, p. 41.

58. Ibid., pp. 50-51 and 48-50.

59. Ibid., pp. 43-48. Cf. Gerlind Werner, Nützliche Anweisung zur Zeichenkunst. Illustrierter Lehr und Vorlagenbücher aus den Beständen der Bibliothek des Germanischen Nationalmuseums. (Ausstellung der Bibliothek der Germanischen Nationalmuseums 21.Juni-7. September,Nürnberg, 1980), Nürnberg: Kuch-Druck, 1980.

60. E.g., Robert Dudley.

61. Soden Smith, as in note 57 above, p. 51.

62. G.-P. Lomazzo, as in I.1 note 36.

63. Cf. Roger De Piles, Principles of Painting, London: J. Osborn, 1743, p. 294 where he lists expression, colouring, design and composition.

64. Felix Bracquemond, Du dessein et de la couleur, Paris: G. Charpentier et Cie., Editeurs, 1895, p. 283:

Table: Le dessin, la couleur; Chaleur, froideur; Dessinateur, coloriste; Reflet, Clair-obscur; Valeur; Trait, model‚ Ligne, masse, silhouette; Perspective, Effet, Execution Ornement, La nature.., Le modelé, l'art et la physique.

65. See the catalogue by Werner, cited in note 59.

66. Samuel van Hoogstraeten, Inleyding tot de hooghe schoole der schilderkonst, Rotterdam: Francois van Hoogstraeten, 1678:

1. Euterpe De Redewikster

9. Urania De Hemelheffster

67. For an introduction to this complex debate see Georg Kauffmann, Poussin-Studien, Berlin: De Gruyter, 1960.

68. John Locke, Some Thoughts Concerning Education, London: Printed for A. and J. Churchill, 1693.

69. Jean Jacques Rousseau, Emile; ou, de l'éducation, The Hague [i.e., Paris]: Jean Neaulme, 1762, 4 vol.

70. For an introduction to these problems see: Theodor Wunderlich, Illustrierter Grundriss der geschichtlichen Entwicklung des Unterrichts im Freien Zeichnen, Stuttgart: W. Effenberger, 1892 and D. Lako, Overzicht van de geschiedenis van het teeken onderwijs in Nederland, Tiel: D. Mijs, 1899. Cf. Ben Schasfoort, Bibliographie van Nederlandstalige literatuur...met betrekking tot tekenonderwijs, Enschede: Stichting von der Leerplanontwikkeling, 1986.

71. See also Oskar Pupikofer, Geschichte des Freihandzeichen-Unterrichts in der Schweiz, St. Gall: Druck der M. Kälin'schen Buchdruckerei, 1890.

72. Cf. A. Fr. Herbold, Die Methode des Zeichenunterrichts der Brüder Ferdinand und Alexander Dupuis, Darmstadt: Druck und Verlag von Carl Wilhelm Leske, 1848.

73. Encyclopédie ou dictionnaire raisonné des arts et des métiers, ed. M. Didérot et M. D'Alembert, Paris: Chez Briasson, David...,1751, vol. 1, pp.889-891:

Les draperies, les fleurs, les fruits, tout enfin doit être dessiné, autant qu'on le peut sur le naturel.

74. See Wunderlich, as in note 70, p. 35:

5. Übungen in der perspektivischen Entwicklung.

75. Cf. Lako, as in note 70, p. 63.

76. Ibid., pp. 68-69.

77. Cf. Wunderlich, as in note 70, p. 92:

a) Zeichnen menschlicher Kopfe (la figure), zuerst nach Gipsmodellen, dann nach lebenden Modellen;

b) Zeichnen ganzer menschlicher Figuren (l'académie), nach Gipsmodellen;

Reihe künstlicher, dann nach natürlichen Blumen.

78. See Lako, as in note 70, p. 97:

IX. De schaduwleer.

79. Larousse, Dictionnaire universel du XIX siecle, as in note 52, p. 59:

Dessiner, a-t-il dit, n'est pas reproduire un objet tel qu'il est; ceci est la besogne du sculpteur, mais tel qu'il parait, et ceci est celle du dessinateur et du peintre; ce dernier achève au moyen de la d‚gradation des teintes ce que l'autre a commencé au moyen de la juste disposition des lignes, c'est la perspective, en un mot, qu'il faut mettre non pas dans l'esprit, mais dans l'oeil de l'élève. Vous ne m'apprenez, dirai-je au maître, avec vos proportions exactes et votre perspective par A plus B, que des vérités, et dans l'art tout est mensonge: ce qui est long doit paraître court, ce qui est courbe paraîtra droit, et réciproquement. Qu'est-ce, en definitive, que la peinture dans sa définition la plus littérale? L'imitation de la saillie sur une surface plane. Avant de faire de la poésie avec la peinture, il faut avoir appris à faire venir les objets en avant. Il a fallu des siècles pour en arriver là. On a commencé par un trait sec et aride, on a fini par les merveilles de Rubens et du Titien, dans lesquelles les parties saillantes comme les simples contours, prononc‚s chacun dans la measure convenable, sont arrivés à cacher l'art tout à fait, à force d'art: voilà le nec plus ultra, voilà le prodige, et ce prodige est le fruit de l'illusion. Quel que soit l'object qu'il se propose de réproduire, le dessinateur est donc tenu avant tout de connaitre, de respecter les lois de la perspective. Il y a la perspective linéaire et la perspective aérienne: la première, dont la géométrie descriptive fournit les regles, suffit au dessin ou il n'est fait usage que de traits, de contours; la seconde, qui a pour objet les modifications apparentes que font subir aux formes les jeux de la lumière et de l'ombre, trouve son application dans les images colori‚es, soit monochromes (comme sont les dessins à la sepia, à l'encre de Chine), soit multicolores. On peut dire que, dans un tableau, le dessin donne la perspective lin‚aire, et la couleur en perspective a‚rienne. Le dessin et la couleur sont donc indispensables, l'un et l'autre, à la peinture. Aussi ne comprenons-nous guère les interminables disputes qui se sont elev‚es sur la question de savoir auquel de ces deux moyens de l'art il convient d'accorder la pre-éminence. Vouloir résoudre cette question, a dit Wattelet, c'est la meme chose que vouloir décider si c'est l'âme ou le corps de l'homme qui constitue la partie la plus éssentielle à son existence.

80. Dictionnaire de pédagogie, as in I.3, note 55, p. 576:

La théorie développée de la perspective fait connaître les procédés imaginé par les géomètres pour obtenir des apparences exactes dans toutes sortes de cas ou il est necéssaire, en effet, pour un peintre de profession, et surtout pour un architecte, d'en savoir executer de telles.

81. Ibid., p. 579:

Le terme de l'étude est le dessin d'après nature. Mais c'est ou commence l'enseignement approfondi. Dans un enseignement élémentaire on ne dépassera pas le dessin d'après la bosse.

82. Ibid, p.580:

Dessin reproduisant des motifs de décoration de surfaces planes ou d'un faible relief: carrelages, parquetages, vitraux, panneaux, plafonds. Lavis a l'encre de Chine et à la couleur de quelques uns de ces dessins.

Releveé avec cotes, et représentation géométrale au traité de solides géométriques et d'objets simples, tels: assemblages de charpente et de menuiserie, dispositions extérieures d'appareils de pierre de taille, grosses pieces de serrurerie, meubles les plus ordinaires, etc. Emploi du lavis pour exprimer la nature des materiaux. Lavis des plans et des cartes.

Ibid., p.575:

L'étude du dessin ne devait pas conduire seulement une partie de ceux qui s'y adonnerait à acquérir de représenter les formes des choses visibles, soit par une pure imitation, soit en imaginant et inventant, et en s'élevant ainsi jusqu'à l'art, mais que ceux qui n'arriveraient pas à acquérir ce talent ou qui ne l'acquerraient que dans une faible mesure, cette étude, si on la fondait sur l'imitation d'excéllents modèles, leur apprendrait à apprecier ce qui est, en fait de choses visibles, exacts ou inéxact, correct ou incorrect, surtout beau ou laid, gracieux ou disgraciuex, s‚ant ou mal séant; qu'elle enseignerait ainsi à mieux voir et a mieux juger, qu'elle formerait enfin, l'oeil et le goût, dont l'utilit‚ est presque universelle.

84. Louis Cloquet, Nouveau traité élémentaire de perspective, Paris: Bachelier, 1823, pp. vi-viii:

Il m'est arrivée plusieurs fois de demander à des artistes qui m'avaient témoigné leur désir de savoir la Perspective, s'ils avaient quelque teinture des premiers élémens de la Géométrie. Ils m'ont répondu negativement, disant d'ailleurs qu'ils n'en avaient pas besoin, qu'ils ne voulaient savoir que la Perspective seule, ou plutot que la Perspective des peintres. Et cependant il n'y a qu'une seule et unique Perspective, et autant vaudrait-il, à mon avis, en manifestant un pareil désir, demander écrire sans vouloir apprendre a lire. Ma manière de raisonner les a eu bientot convaincus, mais ne les a pas tous persuadés. J'ai toujours observé‚ que ceux qui s'en sont rapportés à moi, que ceux qui ont bien voulu se résigner à étudier les principes préliminaires, ont appris très facilement et véritablement la Perspective, science qu'aucun peintre ne doit ignorer. C'est là ce qui m'a engagé à réduire les élémens de la Perspective à ses principes éssentiels, et a les mettre à la portée des lecteurs qui n'ont aucune notion des Mathématiques, ce qui est assez rare heureusement. J'ai donc divis‚ cet Ouvrage en cinq parties. La première traite de la Géométrie élémentaire, dont j'ai cru devoir retrancher toutes les choses étrangères à notre objet, telles que celles qui concernent la mesure des surfaces, des solides, etc. La deuxième contient les principes purement élémentaires de la Géométrie descriptive, qui n'est elle-meme qu'une suite ou une application des principes de la première partie. La troisième traite seulement de la partie de l'Optique qui a un rapport direct à notre objet, c'est à'dire de l'Optique considérée plutot sous la rapport de la Peinture que sous celui de la Physique. La quatrième contient les règles de la Projection des Ombres, très necéssaires, non-seulement aux Peintres, mais encore aux Architectes, aux Dessinateurs, etc. Enfin, la cinquième traite de la Perspective, qui est le dernier et principal objet que nous nous sommes propos‚ de traiter.

Il n'existe, à ma connaissance, aucun ouvrage rédigé sur ce plan.

85. A.W. Cunningham, Notes on the History, Methods and Technological Importance of Descriptive Geometry, Edinburgh: 1868. For a brief discussion of this see P.J. Booker, as in I.3, note 46, pp. 130-132. For a standard treatment of the french context see M. Le Comte de Laborde, De l'union des arts et de l'industrie, Paris: Imprimerie impériale, 1866, 2 vol.

86. Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen, Erlangen: A. Deichert; Cf. David Hilbert, Anschauliche Geometrie, Dover: New York, 1952.

87. K. Lother Wolf and Robert Wolff, Symmetrie Versuch einer Anweisung zu gestalthaften Sehen und sinnvollem Gestalten systematisch dargestellt, Münster: Böhlau Verlag, 1956, 2 vol.

88. H. M. S. Coxeter and S. L. Greitzer, Geometry Revisted, Washington: Mathematical Association of America, 1967, particularly pp. 100-101.

89. H. M. S. Coxeter, Introduction to Geometry, New York: John Wiley, 2nd edition, 1969, p. 175.

90. Margaret Hagen, Varieties of Realism, Cambridge: Cambridge University Press, 1986, p.105.

91. Philip Steadman, Lionel March, The Geometry of Environment, London: Methuen, 1974, p. 25.

92. Ernest R. Weidhaas, Architectural Drafting and Construction, Boston: Allyn and Bacon Inc., 1981, particularly p. 39.

Chapter 5

1. For an introduction to these problems see Pierre Duhem, "Sozein ta phainomena. Essai sur la notion de théorie physique de Platon a Galilee", Annales de philosophie chrétienne, Paris, vol.79/156 (ser.4,VI), 113-138,277-302,352-377,482-514,576-592; translated as: To save the Phenomena: An Essay on the Idea of Physical Theory from Plato to Galileo, Trans. E. Doland and C. Maschler, Chicago: University of Chicago Press, 1969.

2. For a standard work on the astrolabe see: Henri Michel, Traité de l'astrolabe, Paris: Gauthier Villars, 1947.

3. Alfarabi, as in I.1, note 21.

4. Gundissalinus, "De divisione philosphiae," ed. L. Baur, Beiträge zur Geschichte der Philosophie der Mittelalters, Münster, Bd.IV, Heft 2-3, 1903, pp. 112-114.

5. Witelo, Opticae Thesaurus...Vitellonis Thuringopoloni Libri X, Basel: Per Episcopios, 1572 (Cf. New York: Johnson Reprint Corporation, 1972, The Sources of Science, No. 74), p. 217: (Liber Quintus.57):

Possibile est speculum unum planum in camera propria taliter sisti, ut in ipso videantur ea, quae geruntur in domo alia vel in vicis et plateis. Ptolemaeus 7th 2 catoptr.

Sit in camera videntis locus aliquis, in quo existente visu placet videre per speculum planum omne illud, quod alibi agitur: qui locus camerae in quo sistitur centrum visus, sit signatus puncto a: et sit locus, in quo est voluntas aliquid videndi, quod in illo loco agitur, signatus puncto b: sitque rima sive fenestra in camera videntis opposita loco b, que sit g: et ducatur linea bg: & producatur in continuum & directum intra cameram ad aliquam punctum, qui sit d: quod totum postest fieri per astrolabium sive quadrantem vel aliud instrumentum certificationis visuum.

7. Reinerus Gemma Frisius, De radio astronomico et geometro structura, Antwerp: Apud G. Bontium, 1545, 21r:

Omitto autem hic ex proposito, quomodo pictor aliquis insignis imo consistens loco, aut arcem integram, aut aedem sacram, aut civitatem quoque (si velit) ad rationem opticae delineare possit huius nostri radii adminucilo: eo quad haec tum anten narratis, tum ex iis quae dicentur, quivis ingeniosus facile colligem possit. Non possum tamen silentio praeterire summam et facilitatem & commoditatem huius nostri radii, quam architectus aliquis aut pictor adsequi potest, dum stans pede in uno (ut dici solet) totam aedifici faciem sibi opposition in tabulam graphice secundum partium symmetriam depingere possit.

Cf. A. Pogo, "Gemma Frisius, his method of determining differences of longitude by transporting timepieces (1530) and his treatise on triangulation (1533)," Isis, Cambridge Mass., vol. 22, no. 2, February 1933, pp. 467-485.

8. Jacques Bassentin, Amplification de l'usage de l'astrolabe, ed. J. Focard, Paris: Cavellat, 1551, p. 91:

Et pource qu'il n'est pas du tout possible, que le sens et la raison puissent bien connoitre le vraye quantite de l'anglet aigu et variable, par ainsi il seroit tres difficile de naturellement comprendre la certaine quantite d'une chose, par la science de la perspective seulement. A cest cause les anciens geometriciens et mesureurs, ont inventés certains instruments artificiels: et par le moyen d'iceux ont donné facilement a connoitre des quantitez des choses, avec la certitude d'icelle. Mais pour ce qu'il y ha plusieurs et divers instrumens servans & faits pour cest art comme sont un cadran, un triangle geometrique, baculus Jacob, umbraculum visorium, verge geometrique, horloge manuel, quilindre, et autres, desquelz l'usage seroit long a declairer: je passe outre...

10. Manetti, as in I.1 note 31, pp. 44-45.

11. Antonio Averlino detto il Filarete, Trattato di architettura, ed. Anna Maria Finoli e Liliana Grassi, Milan: Edizioni il Polifilo, 1972, p. 653:

guardo uno pavimento che ci sia distesi legni quadri o vuoi guardare uno solare di sotto su: tutte le travature sono equidistanti l'una dall'altra, e sguardando ti parra che sieno e più e meno: secondo chelle ti saranno appresso, ti paranno piu equali, e quantu piu ti si dilungano, tanto piu ti paranno accostate insieme l'una a dosso all'altra, in modo che ti paranno tutt'una. E meglio le vuoi considerare, torrai uno specchio e guarda dentro in esso: vedrai chiaro essere cosi; e se ti fussino al dirimpetto dell'occhio, non ti parebbono se non tutti iguali. E cosi credo che Pippo di ser Brunelleschi fiorentino trovasse il modo di fare questo piano, che veramente fu una sottile e bella cosa che per ragione trovasse che nello specchio ti si dimostra bench‚ coll'occhio ancora, se ben considerrai, tu vedrai quelle mutazioni e diminuzioni.

12. Over one hundred articles have been written on this topic in the past century. For two recent assessments see: Renzo Beltrame "Gli esperimenti prospettici del Brunelleschi,"Accademia nazionale dei Lincei. Rendiconti della classe di scienze morali, storiche e filologiche, Rome, series B, vol. 28, fasc.3, March, 1973, pp.417-468 and Martin Kemp,"Science, non science and nonsense: the interpretation of Brunelleschi's perspective", Art History, London, vol.1, 1978, pp.134-161.

13. Simon Stevin, Derde stuck der wisconstighe ghedaechtnissen van de deusichtighe, Leiden: Ian Bonwensz, 1605 in: The Principal Works of Simon Stevin, ed. D. J. Struik, Amsterdam: N.V. Swetz & Zeitlinger, 1958, vol. II.13, pp. 959-961.

14. Leon Battista Alberti, as in I.i, note 13, pp.56-57.

15. Cf. the author's: Military Surveying and Topography, as in I.1 note 31.

16. For a discussion of these see the author's work on Leonardo, as in I.1, note 14, pp. 108-109

17. David Hilbert, Anschauliche Geometrie, Dover: New York, 1944; translated as: Geometry and the Imagination, New York: Chelsea Publ. Co., 1952.

18. See, for instance: G. Pauschmann, "Zur Geschichte der linsenlosen Abbildung," Archiv für Geschichte der der Mathematik, Naturwissenschaften und der Technik, Leipzig, Bd. 9, 1922, 86-103.

19."Guillaume de St. Cloud, Astronome," Histoire littéraire de la France, Paris, tom. XXV, 1869, p. 73.

20. Re: Levi ben Gerson Cf. Maximilian Curtze, "Die Dunkelkammer, Eine Untersuchung über die Vorgeschichte derselben," Himmel und Erde, Berlin, Jg. XIII, 1901,pp. 226- 232. See also that author's: "Die Abhandlung des Levi ben Gerson über Trigonometrie und den Jacobstab", Bibliotheca Mathematica, Stockholm, N. F., Bd. 12, no. 4, 1898, pp. 97-112.

21. See the author's "Leonardo and the camera obscura" in: Studi Vinciani in memoria di Nando de Toni. Brescia: Ateneo di scienze lettere ed arti. Centro ricerche Leonardiane, 1986, pp. 81-92.

22. Vitruvius, De architectura, ed. Cesare Cesariano, Como: 1521, fol. xxiii:

Excellentemente tange una pulcherima ratione de optica quale fu experta et verificata dal Monastico Architecto Don Papnutio de Sancto Benedicto: si concavo al torno farai un circolo in qualche assicula di quantitate di uncie quatro vel sei, il concavo uncie due vel circa: et questa habia nel centro del concavo uno parvo et brevissimo spectaculo seu foramine quod scopus etiam dicitur: et infixo concordantemente in una valve seu anta di qualche fenestre clause per tal modo in lo loco dove sei non possa introire altra luce: et habi uno pocho di biancho papero vel altra cosa che recipia suso quello che si representera du epso in sino in tuta la terra et coelo sono contenuto.

23. See John Hammond, The Camera Obscura. A Chronicle, Bristol: Adam Hilger Ltd., 1981 for a general treatment. For a concise article with citations from sources see J. Waterhouse, "Notes on the early history of the camera obscura," Photographic Journal, London, vol. 25, 1901, pp. 270-290.

24. Anonymous, Unterweisung im Landschaftmalen und Prospektzeichnen nebst den Hauptregeln der menschlichen Theile, Nurnberg: Verlag der Raspeschen Buchhandlung, 1796, p. 9:

Es glauben wohl einige, dass man durch die Camera obscura eine Gegend am richtigsten aufnehmen kan und es ist wohl wahr, ich bekomme die Gegend richtig, aber es ist viel schöner und besser, wann ich nach der Natur selbst und ohne Camera obscura zeichne.

Die Camera obscura ist hauptsächlich ein gutes Mittel fur Liebhaber, und solche, die nicht nach der Natur zeichnen Können, oder es lernen wollen.

25. Luca Pacioli, Summa di arithmetica, geometria, proportioni e proportionalità, Venice: Paganinus de Paganinis, 1494, Preface:

La perspectiva se ben si guarda senca dubio nulla sarebbe se queste non li se accomodasse. Cioe apieno dimostra el monarcha ali tempi nostri de la pictura maestro Pietro di franceschi nostro conterraneo... Cioe qui in vinegia Gentil e Giovan bellini carnal fratelli. E in perspectivo desegno Hyeronimo Malatini. E in Fiorenza Alexandro Boticelli, Philippino e Domenico grilandaio. E in peroscia Pietro ditto el perusino. E in Cortona Luca del nostro Maestro Piero degno discipul. E in Mantua Andrea Mantegna. E in Furli Melocco con suo caro alievo, Marco Palmezzano. Quali sempre con libella e circino lor opera proportionando a perfection mirabile conducano. In modo che non humane ma divine negliochi nostri sapresentano....

28. E.g., Leonardo da Vinci, A 106v (BN 2038 26v, 1492). Cf. the author's Leonardo Studies I, as in I.1, note 14, p.109.

29. Vitruvius, Architettura con il suo commento, ed. M. Gianbatista Caporali, Perugia: Iano Biganzzini, 1536, fol. 6r:

Hora sia il sesto stato trovato da chi el sia che piu necessario e stato alli mesuramenti di geometria, & prospettiva che a qualunque altro istrumento sia: perche con essu si mesurano tutte be liniali dimostrationi e le angularie cose alle quale si expetta le terminationi delle linie e divisioni...e tanto piu quanto e maggiore la multiplicatione per la inequalita de punti come e notissimo alli experti liniatori, di che specialmente sono di prospettiva.

30. On the problem of universal measurement see the author's "Measurement, quantification and science": L'époque de la renaissance IV (1560-1600), ed. Tibor Klaniczay, Budapest: Akademiai Kiado, 199?.

31. On Fabrizio Mordente see: Paul Lawrence Rose, "The Origins of the proportional compass from Mordente to Galileo," Physis, Florence, anno X, fasc. 1, 1968, pp. 53-69.

32. Fabrizio Mordente, Il compasso del Signor Fabritio Mordente con altri istromenti mathematici ritrovati da Gasparo suo fratello, Antwerp: apresso Cristofano Plantino, 1584 (Cf. manuscript copy, Munich, Bayerische Staatsbibliothek, Cod. it. 11).

33. For a survey of these problems and a brief history of the proportional compass or sector see: Ivo Schneider, Der Proportionalzirkel. Ein universelles Analogrecheninstrument der Vergangenheit, Munich: R. Oldenbourg Verlag 1970, (Deutsches Museum. Abhandlungen und Berichte, 38 Jg., 1970, Heft 2).

34. On the question of gauging see Menso Folkerts, "Die Entwicklung und Bedeutung der Visierkunst als Beispiel der praktischen Mathematik der frhen Neuzeit, "Humanismus und Technik, Berlin, Band 18, Heft 1, 31 Mai 1974, pp. 1-41. Cf. Grete Leibowitz, Die Visierkunst in Mittelalter, Phil. Diss, Heidelberg, 1933.

35. On Schissler see the excellent book by Maximilian Bobinger, as in chapter 2, note 40.

36. This will be the subject of another study by the author. Here by way of introduction some preliminary facts may be noted. A compass by Coignet, in the collection of Commmandant Rasquin (Antwerp) contains the following lines:

segmentorum globi

The Newberry Library manuscript attributed to Lencker contains the lines:

Reductio corporum

A sixteenth century German manuscript in Munich, Bayerische Staatsbibliothek, Germ. N.4154, contains 8 different lines. A Manuscript by Zugmesser, with whose colleague Galileo quarreled, now in Stuttgart, Landesbibliothek, HB XI 19, contains 15 lines.

37. Hans Lencker [attributed], Perspectiva, Chicago, John M. Wing Foundation, Newberry Library, Ms.B.128.

38. See, for instance, Ludolf von Mackensen, Die erste Sternwarte Europas mit ihren Instrumenten und Uhren - 400 Jahre Jost Bürgi in Kassell, Munich: Callwey, 1982.

39. Lencker, as in note 37 above.

40. See Stillman Drake, "Tartaglia's Squadra e Galileo's compasso," Annali dell'istituto e museo di storia della scienza di Firenze, Florence, anno II, fasc. 1, 1977, pp. 35-54. Also his "Galileo gleanings IX. An unrecorded manuscript copy of Galileo's Use of the Compass," Isis, Cambridge, Mass., vol. 51, 1960, pp. 56-63. Cf. The introduction to his translation of Galileo Galilei, Operations of the geometric and military compass, Washington: Smithsonian Institute, 1978.

41. Re: Michel Coignet see: G. Enestrom, "Uber den Partometer von Michel Coignet," Bibliotheca Matematica, Stockholm, 3 Folge, Band 7, Heft 4, p.397.

42. Ladislao Reti, "Elements of machines," in: L. Reti, ed., The Unknown Leonardo, London: Hutchinson and Co., 1974, pp. 272-273. For an early seventeenth century expression of these connections see Blancanus, as in chapter 4, note 11, when speaking of the six speculative mathematical [sciences], p.389:

Quarta, mechanica, quae de machinis agit, sive, ut ait Aristoteles versatur circa artificata, sicuti naturalia de sex machinis praecipuis, libra, vecte, trochlea, axe in peritrochio, cuneo, cochlea egregia demonstrat: et quia in eis considerat quantitate virium moventium, ponderum motuum, temporum, quibus moventur, & machinas ipsas tanquam lineas quasdam circa centra revolutas, ideo geometrae subalternatur, idest, geometrice demonstrat. Eius pars subtilissima est, quae centra gravitatis in planis, ac solidis perscrutatur.

43. On the question of the four powers see: Kenneth D. Keele, Leonardo da Vinci's Elements of the Science of Man, New York: Academic Press, 1983, particularly, pp. 93-130.

44. Leonardo da Vinci, W19070v (K/P 113r):

Il libro della scientia delle macchine va inanzi al libro de govamenti.

45. Leonardo da Vinci, K 49 [48 et 15]r:

La proportione non solamente nelle numeri e misure fia ritrovata ma etiam nelli suoni, pesi, tempi essiti ecqualunche potentia sicia.

46. Leonardo da Vinci, Ca 203va:

Ma direno solamente i moti essere di 2 nature, delle quali l'uno e materiale e l'altro spirituale, perche non e compreso dal senso del vedere, overo direno d'uno essere visibile e l'altro invisibile.

47. See, for instance, CA 66rb (186r, c.1505):

Il notare sopra dell'acqua insegna alli omini come fanno li uccelli sopra dell'aria.

48. For a full analysis of percussion in relation to Leonardo's physics see the author's Leonardo da Vinci Studies II: 1.2

49. For an analysis of these passages see the author's Leonardo da Vinci Studies III.

50. Hence, on CA 252rb (681r, c.1490-1492), he speaks, for instance of: "questa terrestre e mundiale macchina".

51. E. J. Dijksterhuis, The Mechanization of the World Picture, trans. C. Dikshoorn, London: Oxford University Press, 1961.

52. On the general mathematical context see: Paul Lawrence Rose, The Italian renaissance of mathematics, Geneva: Librairie Droz, 1976 (Travaux d'humanisme et renaissance, CXLV).

53. See: Giuseppe Boffito, Il primo compasso proporzionale costruito da Fabrizio Mordente e la operatione cilindri di Paolo dell'Abbaco, Florence: Seeber, 1931. Cf. Maria Luisa Bonelli, "Di una bellissima edizione di Fabrizio Mordente...", Physis, Florence, vol. 1, 1959, 127-148, particularly p.144.

54. On Egnazio Danti see Thomas Settle, "Egnazio Danti's great astronomical quadrant," Annali dell'istituto e museo di storia della scienza di Firenze, Florence, anno IV, fasc. 2, 1979, pp. 3-13. Cf. Thomas Settle, "Ostilio Ricci. A Bridge between Alberti and Galileo," XII congrès internationale d'histoire des sciences, Paris, tome 3 B, 1971, pp. 117-122; also his "The Tartaglia Ricci problem; towards a study of the technical professional in the 16th century." Atti del convegno internazionale di studio. Giovanni Battista Benedetti e il suo tempo. Istituto veneto di scienze lettere ed arte, Venice, 1987, pp. 217-226.

55. Giorgio Vasari, Jr., Raccolta fatto dal Cav. Giorgio Vasari di varii instrumentii per misurare con la vista, 1600, Florence, Biblioteca Riccardiana, Ms. 2138. Cf. Loredana Olivato, "Profilo di Giorgio Vasari il giovane," Rivista dell'istituto nazionale d'archeologia e storia dell'arte, Rome, n. s. anno 16-17, 1969-1970, pp. 181-229.

56. On Ursus see Nick Jardine, The Birth of History and Philosophy of Science. Kepler's A Defence of Tycho against Ursus...,Cambridge: Cambridge Univesity Press, 1984.

57. Cf. Ernst Zinner, Deutsche und niederlandische astronomische Instrumente des 11-18 Jhdt, Munich: Beck Verlag, 1956.

58. Andreas Albrecht, Abriss und Beschreiburg eines sonderbaren nützlich notwendigen Mechanischen Instruments, Nurnberg: Ieremiam Dumlerr, p.34:

Wie mit diesem Instrument ein Gebau, ein Landschaft oder andere corperliche vor Augen stehende Ding perspectivich genommen und aufgerissen werden sollen; p. 36: Zum Beschluss könnt ihr mit diesem Instrument auch alle geometrische und perspectivisch Figuren in was Gröss ihr wolt nur dass ihr die verjüngte Maass nach eurem Gefallen verandert, verkleinern und vergrössern.

59. Leonardo da Vinci, Codice Atlantico, 190ra.

60. See, for instance, Codice Atlantico, 112ra, 251rb, BM 104ra. All the passages involved have been considered in volume three of the author's Leonardo Studies.

61. For a conservative assessment of these passages see Albert van Helden, "The invention of the telescope," Transactions of the American Philosophical Society, Philadelphia, vol. 67, part 4, 1977, pp. 1-67.

62. E.g. Ernst Cassirer, Substance and Function, trans. William Curtis Swabey and Marie Collins Swabey, Chicago: Open Court Publishres, 1923; New York, Dover Publications, 1953.

63. Donald P. Greenberg, "Computer graphics in architecture," Scientific American, New York, Vol. 230, May 1974, pp. 98-106. A basic text is William M. Newman, Robert F. Sproull, Principles of Interactive Computer Graphics, Tokyo: McGraw Hill, Koga Kusha Ltd., 1973.

64. R.A. Reynolds, Computing for Architects, London: Butterworths, 1987, particularly p.64

65. Alan Pipes, ed., Computer Aided Architectural Design Features, London: Butterworths, 1986, particularly fig. 5.2.

66. Sherley Werner Morgan, Architectural Drawing, Perspective, Light and Shadow, Rendering, New York, 1950, p.

Chapter 6

1. Vasari, as in I.2, note 1, vol.II, p.90.

2. Ibid., vol.I., p.233.

3. Ibid., vol.I, p.328; vol.II, pp.90-230; vol.III, pp.41, 55, 133; vol.IV, pp.60, 195.

4. Benedetto Dei, Cronaca Fiorentina dal 9 dicembre 1430 al 1480, Florence, Archivio di stato, manoscritti no. 119, fol. 32v:

Florentia bella a 66 botteghe di speziali e a 84 botteghe di legnaiuoli di tarssie e intagliatori.

Cf. his Memorie, Florence, Biblioteca Riccardiana, Cod. 1853, fol. 90r.

5. Vasari, as in I.2, note 1, vol.I, p.251.

6. Ibid., vol.I, p.308.

7. Ibid., vol.I, p.237.

8. Erich Auerbach, Mimesis, The Representation of Reality in Western Literature, trans. Willard R. Trask, Princeton: Princeton University Press, 1945.

9. See, for instance, Hans Schüritz, Die Perspektive in der Kunst Albrecht Dürers, Frankfurt: Heinrich Keller, 1919. Cf. Karl Rapke, Die Perspektive und Architektur auf des Dürerschen Handzeichnungen, Strasbourg: J.H. Ed. Heitz, 1902.

10. Vasari, as in I.2, note 1, vol.I, pp.232-233.

11. Sir E. H. Gombrich, Means and Ends, as in chapter 1, note 30.

12. Cf. Franz Boas, Primitive Art, New York: Dover Publications, 1955. On the problem of primitive mentality see: Ernst E. Baesch, Das Magische und das Sch"ne. Zur Symbolik von Objekten und Handlungen, Stuttgart: Froomann-Holzboog, 1983.

13. Sir E. H. Gombrich, The Sense of Order, London: Phaidon, 1979.

14. Cf. C. Van der Sleyen, Das alte Ägypten, Frankfurt: Verlag Ullstein, 1975, pl. 46, (Propyläen Kunstgeschichte, Bd. 15).

15. Sir E. H. Gombrich, Art and Illusion, Princeton: Princeton University Press, 1960, p.129.

16. Ibid., p. 131.

17. Cf. Heinrich Schäfer, Principles of Egyptian Art, tr. John Baines, Oxford: Clarendon Press, 1974.

18. Pliny, Natural History, trans. H. Rackham, Cambridge, Mass: Harvard University Press, vol. IX, 1968, pp. 306-311 (XXXV.65-66).

19. Ibid., vol. IX, 1968, pp. 176-177 (XXIV,XIX.65):

vulgoque dicebat ab illis factos quales essent homines, a se quales viderentur esse.

20. One of the fundamental contributions of Sir Ernst Gombrich has been to demonstrate that psychological projections continue to imbue art. See: Art and Illusion. Princeton: Princeton University Press, 1960.

21. An image in the mind which is visual cannot be recorded or measured. It needs a verbal description. Hence, although visual in the mind, it nonetheless needs a verbal filter before it can be communicated.

22. Vasari, as in I.2, note 1, vol.I, p.209.

23. Auerbach, as in II.1, note 8.

24. Cf. Sir E. H. Gombrich, "Illusion and art," in: Illusion in Nature and Art, ed. R.L. Gregory and E.H. Gombrich, London; Duckworth, 1973, pp. 193-243, particularly pp. 230-231.

25. Michael Kubovy, The Psychology of Perspective and Renaissance Art, Cambridge: Cambridge University Press, 1986, pp.52-64.

26. For an analysis see Marisa Dalai Emiliani, "Il ciclo del Foppa nella cappella Portinari."

27. Alessandro Parronchi, Studi su la dolce prospettiva, as in I.2, note 6, pp.340-348.

28. Alessandro Parronchi, Masaccio, Florence: Sadea Sansoni, 1966. His reconstruction is reproduced in: L'opera completa di Masaccio, ed. Paolo Volponi, Luciano Berti, Milano: Rizzoli, 1968, p.96.

29. Cf. Jean Seznec, The Survival of the Pagan Gods, trans. Barbara F. Sessions, New York: Harper and Row, 1953.

30. For an analysis of the contents of these paintings, cf. S. J. Freedberg, Painting in the High Renaissance in Rome and Florence, Cambridge, Mass.: Harvard University Press, 1961, 2 vol.

31. Leonardo da Vinci, A 96r (BN 2038 16r, TPL 119, 1492):

Ti rispondo che tu debbi porre il primo piano col punto al'altezza de l'ochio de riguardatori d'essa storia et in sul detto piano figura la prima storia grande e poi diminuendo di mano in mano le figure e casamenti in su diverse colli e pianure farai tutto il fornimento d'essa storia.

For another discussion of this passage see: Sir E. H. Gombrich, Means and Ends, as in I.1, note 30, p. 10.

32. Gotthold Ephraim Lessing, Läokoon, oder über die Grenzen der Malerei und Poesie, Stuttgart: Reclam, 1964, (271/71 a/b), p. 114:

Ich schliesse so. Wenn es wahr ist, dass die Malerei zu ihren Nachahmungen ganz andere Mittel, oder Zeichen gebrauchet, als die Poesie; jene nämlich Figuren und Farben in dem Raume, diese aber artikulierte Tö