Dr. Kim H. Veltman
IV Classification
1. Introduction
2.
Optics
3.
Architecture
4.
Drawing
5.
Drawing
Education
6. Mathematics
7.
Conclusions
From the outset, the classification of perspective posed problems. Etymologically it was linked with optics, which could not be classed simply. Historically it was linked with architecture, which equally eluded simple classification. By the nineteenth century the nature of the problem began to come into focus. Systems of classification had forged clear distinctions between (subjective) art and (objective) science, as well as between (theoretical) science and (practical) technology. Perspective had the embarrassing characteristic of belonging clearly to all of these. It belonged to art, because it created spatial effects in paintings and it was accordingly classed under drawing. At the same time its theoretical principles were so clearly connected with the mathematical projections underlying scientific demonstrations, that it came to be listed under descriptive geometry. Finally, it also involved instruments, such that it was also classed under technology. These three headings, plus the historical one connecting it with architecture, have been maintained in the Library of Congress classification to this day.
On the surface, this was merely a matter of semantics, a search for convenient cubbyholes in the case of a borderline subject. There were, however, dramatic consequences. It meant, for example, that numerous innovations pertaining to perspective accrued to the larger categories to the extent that, when the artistic aspects of perspective came under fire, it could reasonably appear to some that perspective had died although the number of treatises on perspective in these other fields continued to grow. The continued association with optics perpetuated a confusion between subjective optical questions of how an object is seen, and objective geometrical questions of an object's projections. The connection with drawing meant that perspective became entangled in conflicting philosophies of artistic education and was affected by changing fashions therein.
Ironically, the connection with mathematics, which seemed
the most obvious of all, proved to be the most perplexing. By the mid-sixteenth
century mathematicians such as Commandino had established the mathematical
foundations of perspective. These were further clarified by Guidobaldo del Monte
and Stevin, and consolidated by Desargues. In the nineteenth century, with the
development of descriptive geometry, it seemed obvious to class perspective as a
branch thereof. In the twentieth century, with the development of algebraic
geometry, this more general concept embraced both descriptive geometry and
perspective. The problem was that algebraic geometry was non-visual, which posed
a curious paradox: perspective, the chief means for visualizing space and
objects, was classed as a subset of a non-visual method. Meanwhile, where
perspective stands in relation to other branches of projection such as isometry,
affinity and topology has become an open question. In order to gain further
insight into these problems of classification it will be useful to consider each
of them in turn.
Etymologically, both the term for linear perspective and for optics stemmed from the Latin, perspectiva.1 Already in Antiquity, optics had eluded attempts at simple classification. The ancients had partially sidestepped this problem by dealing with its geometrical, philosophical and physical aspects separately. Aristotle had included optics among the mixed sciences,2 which subsequently led to its being classed among the mechanical arts by some mediaeval thinkers.3 Considered less important than astronomy and music, optics was excluded from the quadrivium section of the mediaeval seven liberal4 arts and yet, ironically, by the late thirteenth century, it emerged as chief of the mathematical sciences. In the words of Peckham, archbishop of Canterbury:
Among the investigations of physics, light is most pleasing to students of the subject. Among the glories of mathematics it is certitude of demonstration that most highly exalts the investigators. Therefore, optics (perspectiva), in which demonstrations are devised through the use of radiant lines, and in which glory is found physically, as well as mathematically, so that optics (perspectiva) is adorned by the flowers of both, is properly preferred to [all] the teachings of mankind.5
The Renaissance continued to class optics as a branch of mathematics. This continued into the seventeenth century when two trends emerged: one listing optics as merely another item in a growing list of mathematical sciences, the other distinguishing between optical, mechanical, geographical and other sciences. Some examples from both traditions will be needed to reveal the constantly shifting interplay between optics and perspective. The idea of listing disciplines grew out of the mediaeval tradition of the seven liberal arts, by means of which optics was subsumed under geometry as one of the basic disciplines. In Gregor Reisch's Philosophical pearl (1517), for instance, this list included: grammar, dialectic, rhetoric, arithmetic, music, geometry, astronomy, natural philosophy, alchemy, vegetative and sensitive soul, rational soul and moral philosophy.6 In the works of Joachim Fortius (1531), this list was reduced to grammar, dialectic, rhetoric, mathematics and divination, but the branches of mathematics were correspondingly expanded to include the sphere, astronomy, cosmography, book of time, table of time, perspective (optice) and mathematical chaos (which included practical geometry, surveying, cosmography, astrology, meteorology and dreams).7 John Dee's (1570) list, in his mathematical preface, included optics, amidst a number of more exotic branches: "perspective [i.e. optics], astronomie, musike, cosmographie, astrologie, statike, anthropographie, trochilike, helicosophie, pneumatithmie, menadrie, hypogeiodie, hydragogie, horometrie, zographie [i.e. perspective], architecture, navigation, thaumatargike, archemastrie."8
In the seventeenth century, these lists continued to change, with optics emerging increasingly as an independent category. Valentin Andreae (1614), for instance, included geometry, arithmetic, statics, astronomy, gnomonics, automata, optics, architecture, fortification, surveying, and polyhedra.9 Robert Fludd's (1617) list included arithmetic, music, geometry, optics, pictorial art, military art, science of motion, science of time, cosmography, astrology and geomancy.10 Blancanus (1620) listed only six branches of speculative mathematics: geometry, arithmetic, optics, mechanics, music and astronomy,11 to which he added six others which were their practical equivalents.12 Ciermans (1641) also chose a dozen disciplines in order to have them correspond with the twelve months: geometry, arithmetic, optics, statics, hydrostatics, nautical science, architecture, war, military machines, geography, astronomy and chronology.13
Schwenter (1651) increased the list to sixteen: arithmetic, geometry, stereometry, music, optics and perspective, catoptrics, astronomy and astrology, gnomonics and clocks, statics, motion, pyrotechnics, pneumatics, hydraulics, writing, architecture and chemistry.14 Four decades later, with the Jesuit, Milliet de Chales (1690), optics had moved to twentieth place in a list of thirty-one treatises on: the fourteen books of Euclid, the Sphaerics of Theodosius, conic sections, arithmetic, trigonometry, algebra, practical geometry, mechanics, statics, geography, magnetism, civil architecture, art of dyeing, stone cutting, military architecture, hydrostatics, fountains and rivers, hydraulic machines, navigation, optics, perspective, catoptrics, dioptrics, music, pyrotechnics, astrolabes, gnomonics, astronomy, astrology, meteors and calendars.15
When the librarians at Göttingen developed their systematic subject catalogue (1730's ff.) their list for mathematics was effectively a new combination of these disciplines: 1) speculative and practical geometry, arithmetic, algebra; 2) optics, catoptrics, dioptrics; 3) cosmography, on the sphere, globes etc.; 4) astronomy, 5) gnomonics; 6) judiciary astrology; 7) physiognomy, chiromancy and other divinatorial arts; 8) civil architecture; 9) military architecture; 10) military art; 11) art of navigation, 12) hydraulic art and hydrostatics, machines; 13) art of painting and sculpture; 14) art of writing; 15) music, 16) mechanical and illiberal arts.16
While many more lists could be cited the above examples amply confirm that there was no fixed order for listing the mathematical sciences, that optics was sometimes listed as a general category including perspective, and sometimes listed separately from perspective. Similar conditions obtained in another tradition, which arranged the mathematical sciences in groups. For example, Hérigone (1644), in his new course on mathematics, divided his work into five tomes (adding a sixth as a supplement). The first tome dealt with theoretical geometry; a second with arithmetic, computus, algebra and analysis. A third tome contained trigonometry, practical geometry, art of military and mechanical warfare; a fourth contained the doctrine of the sphere and geography while a fifth dealt with optics, catoptrics, dioptrics, perspective, spherical trigonometry, planetary theories, gnomonics and music.17 Ozanam's (1697) five tomes were somewhat differently arranged: 1) introduction to mathematics and Elements of Euclid; 2) arithmetic, trigonometry and sine tables; 3) [practical] geometry; 4) mechanics and optics (perspectiva); 5) geography and gnomonics.18
Wolff (1730-1738) reduced this scheme to four tomes: 1) mathematical method, arithmetic, geometry, trigonometry; 2) mechanics, with statics, hydrostatics, aerometry and hydraulics; 3) optics, perspective, catroptrics, dioptrics, the sphere and spherical trigonometry and astronomy, both spherical and theoretical; 4) geography with hydrography, chronology, gnomonics, pyrotechnics,, military and civil architecture.19 Frobes (1743-1746) preferred to further subdivide Wolff's third to fourth categories thus producing six parts: 1) arithmetic, geometry and trigonometry; 2) mechanics, hydrostatics, aerometry and hydraulics; 3) pyrotechnics as well as military and civil architecture; 4) optics, catoptrics, dioptrics and perspective; 5) astronomy; 6) geography, chronology and gnomonics.20
Clemm (1764) chose to distinguish between five kinds of sciences: arithmetical sciences (computation of numbers, letter computation, practical computation); geometrical sciences (elementary geometry, trigonometry, plane and spherical; practical geometry, higher geometry); static sciences (statics or mechanics, hydrostatics, aerometry, hydraulics), optical sciences (optics, catoptrics, dioptrics, perspective) and astronomical sciences (astronomy, spherical and theoretical geography, chronology, gnomonics, civil and military architecture).21 By the time of Wiegleb's edition of Martius' Instruction in natural magic (1796-1798) this list of different sciences had been transformed into eight realms of artifice (Kunststücke): electricity, magnetism, optics, chemistry, mechanics, computation, economics and cards (sic!).22 Meanwhile, Kästner (1792) reduced the sciences to two basic divisions: 1) mechanical and optical sciences; 2) astronomy, geography, chronology and gnomonics.23 Lorenz (1798) devoted a first book to arithmetic and geometry; a second book to mechanical, optical and astronomical sciences and a third book to general computation of sizes.24 Such examples confirm that even in this second, more organized tradition, there was still no fixed order for listing the mathematical sciences, that optics25 was sometimes listed as a general category including perspective, and sometimes listed separately from perspective.
If there was debate about the relationship of optics and perspective within a general framework of the mathematical sciences, there was an even greater difference of opinion concerning the respective branches thereof. In John Dee's (1570) table, for instance, "perspective" [i.e. optics] was that "which demonstrateth all maners and properties of all radiations direct broken and reflected," while "zographie" [i.e. perspective] was that "which demonstrateth and teacheth, how, the intersection of all visuall pyramids, made by any plaine assigned (the center, distance and lightes being determined), may be, by lines and proper colours represented".26 Another author divided optics into theoretical and practical; dividing the latter into technical, perspectival, planisphaeric and anamorphic; subdividing perspectival in turn into vertical (with its categories united or pictorial and separated or scenical) and horizontal; while subdividing anamorphic into ithyoptic, catoptric and dioptric.27
Guldin (1635) divided optics into optics proper, catoptrics, dioptrics and diocatoptrics; subdividing optics proper into perspectiva [i.e. optics] and perspectiva [i.e. perspective], the latter of which he further divided into orthography, stereography and scenography28 (an adaptation of the Vitruvian scheme, cf above p. ). He also subdivided catoptrics into plane, convex, concave and burning mirrors. Haesel's (1672) scheme bore some resemblance to this. He began by dividing optics into general and specific, the latter of which he subdivided into parts which dealt with 1) straight or direct ray (including ichnography, orthography, scenography and sciography); 2) reflected ray or catoptrics (as in plane, concave and convex mirrors) and 3) a refracted or infracted ray (as in mesoptics through glass, water or air).29
Rather different again was the view of Adelung (1781) who also looked upon perspective as a branch of optics:
Perspective in turn divides itself into [1] linear perspective which, with the help of geometry, teaches the proper foreshortening of straight lines as, for example, the parts of a building; [2] aerial perspective, which falls entirely within the domain of the painter and teaches how to establish light and shade in accordance with changes which colour of the air brings to bodies and their colours at a certain distance and [3] mirror perspective, which teaches one how to draw irregular and distorted figures, which spherical, conic and other mirrors again restore to their regular shape.30
Meanwhile, Ciermanns (1641), subdivided optics into eight categories: perspective, orthography, scenography, practice, compendious [perspective], scenography, curious [perspective] and problematic [perspective].31 Wiegleb (1786-1798) focussed on eight quite different subdivisions: 1) the eye and its imitation; 2) curious perspective [i.e. anamorphosis]; 3) plane mirrors; 4) concave and convex mirrors; 5) prisms and prismatic colours; 6) convex and concave lenses; 7) perspective; 8) instruments and machines for drawing.32
Von Schlosser (1935) assumed that the discovery of perspective led to a clear distinction between optics (perspectiva communis) and perspective (perspectiva artificialis).33 Panofsky (1927) spread this idea.34 However, as the above examples confirm, there was no ready consensus about either the relation of optics and perspective or their further subdivisions. Granted there were cases when linear perspective was distinguished by an extra adjective as in prospectiva pingendi, or prospectiva pratica, but in general, the terms for optics and perspective tended to remain interchangeable. Indeed Leonardo, adopted the title of Peckham's treatise on optics, when he headed a section on perspective in his Treatise on painting: prospettiva comune.35 And Leonardo's contemporary, Cesariano added a basic diagram relating to perspective in a manuscript version of Peckham's text.36 This continued interplay of meanings helps explain the enduring impact of optics on perspective in terms of both titles and content, although it varied from country to country. In Italy, for instance, because the term prospettiva continued to mean both "optics" and "perspective," these connections remained largely implicit. In Germany, works which specifically mentioned optics in their titles, were rare, as, for example, Mühlert's (1821) Art of shadows following optical laws, Öttingen's important article (1906), Judgment of perspective images with references to the viewpoint of the observer and Riegel's (1952) The visual image, Handbook of Perspective.
In France, this connection between optics and perspective, became explicit in the sixteenth century. Hence the Antwerp encyclopaedist Ringelbergius (1531), whose works were published in Paris, had a section on perspective headed optics (Optice), while Androuet Du Cerceau (1551) entitled one of his works Twenty figures for they contain most ancient optical views which they call perspective. In the seventeenth century, Nicéron (1638) wrote Curious perspective or the artificial magic of wonderful effects of optics, catoptrics and dioptrics, and Huret (1670) entitled his treatise on perspective simply, Optics of portraiture. The eighteenth century saw only isolated examples such as La Caille's (1750, etc.) Elementary lessons of optics, which dealt with perspective. In the nineteenth century Vallée's (1844), Theory of the eye played an important role in asserting that the laws of descriptive geometry, perspective and vision coincided. He began his work with a survey of famous contributions to optics by Leonardo da Vinci, Kepler, Soemmering, Brewster and Young, noting that his own interests had begun more than two decades earlier:
...in 1821 we published the Treatise of the science of drawing. In this work we consider painting, drawing in its different genres and in general the art of imitating objects in order to produce a greater or lesser illusion, as an application of the rules by means of which one can deceive the eye. Hence the theory of vision serves as the basis of our treatise, where it is explained in some detail and often envisaged under new aspects. In this work, of which perspective and shadows are important parts, geometry guides us constantly and has led us to an explanation of the achromatic nature of the eye based on the non-homogeneity of the vitreous body.37
The second half of the nineteenth century saw a number of new works, which assumed close links between optics and perspective, such as Babinet's (1855), Studies and readings on the sciences of observation, Charles' (1884), Elementary treatise on linear perspective and perspective of view, Payen's (1884), Extract of a treatise on ocular perspective, and numerous works on perspective of observation including Pierre (1885), Daniel (1891), Tensi (1895), Watelet (1893), Chevrier (1900), Legrand (1903, 1908), Lienaux (19135-1949) and more recently Raynaud (1962).
In the Netherlands emphasis on the optical aspects of perspective began with Vredeman de Vries (1560), who entitled one of his earliest collections of engravings, Twenty most select buildings of scenography or perspective, as buildings summoned forth to the eye are called to the ordinary painter, and later wrote (1604-1605) etc. Perspective, that is, the most famous art of looking in or through the axis of the eyes, a title which recurred in subsequent editions by his student Marolois (1628, 1638) etc. Another of Vredeman de Vries' students, Hondius (1623, etc.), wrote an Instruction of optics or perspective, as if the two terms were fully interchangeable. Similarly, a later edition of Marolois (1653) was entitled simply, Optics or perspective.
In England, the work of Hondius and Marolois was a starting point for a treatise by Moxon (1670): Practical perspective or perspective made easie teaching by opticks. In 1701, Lamy published his Traité de perspective, ou sont contenus les fondemans de la peinture, which was translated into English the following year (1702) as: Treatise of perspective or the art of representing all manner of objects as they appear to the eye in all situations, which almost certainly influenced Brook Taylor (1715, etc.) when he entitled his seminal work: Linear perspective or a new method of representing justly all manner of objects as they appear to the eye in all situations. Meanwhile, Shuttleworth (1707), had written A treatise of optics direct...to which is added an appendix on perspective. Later in the century, Harris (1775), also dealt with perspective in A treatise on optics containing elements of the science. The nineteenth century saw further works linking optics and perspective, such as Keating's (1812) Eidometria or optic mensuration and (corollary) perspective or Burnet's (1837, etc.), An essay on the education of the eye with reference to painting. This tradition continued in the twentieth century with works such as Roberts (1904), Perspective of sight, and Myslak's (1967), Point of view perspective.
These analogies between optics and perspective affected both fields. In the sixteenth century, Leonardo had explored tensions between a spherical--or cylindrical--plane of vision, and the rectilinear plane of perspectival representation. In the seventeenth century, Bosse deliberately contrasted a cylindrical plane of vision, with a rectilinear plane of perspective, to demonstrate why one must not draw what one sees. By the nineteenth century, the development of descriptive geometry encouraged the assumption that there were universal laws of reality, which applied equally to geometry, vision and representation. The window principle now served to demonstrate that the laws of optics and perspective were identical, and became a standard example in introductions to textbooks on perspective, as in Ruskin (1859):
When you begin to read this book, sit down very near the window, and shut the window. I hope the view out of it is pretty; but, whatever the view may be, we shall find enough in it for an illustration of the first principles of perspective (or, literally, of "looking through").Every pane of your window may be considered, if you choose, as a glass picture; and what you see through it, as painted on its surface...
...Every picture drawn in true perspective may be considered as an upright piece of glass, on which the objects seen through it have been thus drawn. Perspective can, therefore, only be quite right, being calculated for one fixed position of the eye of the observer; nor will it ever appear deceptively right unless seen precisely from the point it is calculated for.38
Ruskin quietly relegated to an appendix the problem of visual angles39 (cf. p. ) which implicitly challenged this equation between vision and representation. Cassagne (1879), referred to equally sized objects positioned at different heights strictly in terms of their projections, without references to their different visual angles. For him the visual angle was only of interest as applied to the ensemble, particularly with respect to establishing the proper distance for viewing and drawing objects.40
In the meantime, the initial wave of enthusiasm for descriptive geometry had passed, and thinkers had become aware that there were problems with these easy equations between geometry, perspective and optics. Jules de la Gournerie (1859), for instance, noted that:
Perspective is a graphic art of a special kind. It poses practical difficulties peculiar to it alone and which, in the course of several centuries have occupied a great number of learned men and artists. Authors who have treated it as a simple application of descriptive geometry have not been able to give it the necessary developments. There are moreover grounds to believe that several of them deigned not to study the older treatises. In fact practically all the students of Monge believed that all the graphic arts presented only uncertainty and confusion prior to [the coming] their master. One finds in the writings of several of the most famous of these, assertions which are completely erroneous in this regard.41
At about the same time, the researches of Hering and Helmholtz42 were suggesting that the perceptual space of optics might be non-Euclidean, and hence fundamentally different from the Euclidean space of perspective. This led gradually to a whole body of literature on alternative projection methods designed to approximate more closely the realities of visual perceptions (cf. pp. and ).
When Panofsky wrote his landmark essay on perspective as a symbolic form (1927),43 he assumed that the discovery of linear perspective brought about a change in Euclidean theories of visual angles. This did not occur in the sixteenth century, as he imagined, but rather in the nineteenth, and then only briefly under the impetus of descriptive geometry. And even then, perspective remained under the spectre of being classed as a branch of optics such that whenever there were innovations in optical theory, there were usually corresponding innovations in perspectival methods. This has continued to the present day such that works relevant to perspective are still sometimes classed as optics.
In the case of architecture, this question of classification posed an even greater problem. For in a sense it could be argued that every illustrated architectural book related somehow to perspective. We have already noted some of the chief architectural themes contained in actual treatises on perspective (pp. ). And while it is clearly beyond the scope of this essay to provide a comprehensive survey of all architectural works which might have a bearing on perspective, it will be useful to outline some of the more obvious points of contact. At the most obvious level, there were architectural books with sections on perspective, and some books specifically devoted to architectural perspective. In order to understand the larger context of these problems, a brief excursus into the history of classification of architecture itself will be required. This will lead to a consideration of links with descriptive geometry and specialized categories, such as architectural drawing and drafting. Each of these will be considered in turn.
Just as there were general treatises on perspective with sections on architecture (cf. p. ), so too were there general treatises on architecture with sections on perspective. For instance, Furttenbach's Recreational architecture (1640), contained numerous perspectival views and a special section on stage design. Caramuel de Lobkowitz (1678) offered various perspectival methods of interest to architects. Guarino (1737), included an important discussion of perspective in his Civil architecture, while Amico (1750), in his Practical architect added a "Compendium of perspective." Vittone (1760, 1766) also discussed perspective at some length. This tradition continued into the nineteenth century with general works such as Gwilt's Encyclopaedia of architecture (e.g. 1842, 1851, 1861).
Stonecutting was a branch of architecture, which involved perspective in a special way. During the middle ages knowledge of this trade was kept a secret by the guilds. Roriczer (1486) published on the topic in a secretive way. Philibert de l'Orme (1567), began to explore the relations of stonecutting and perspective in veiled terms. Yet it was not until the 1640's that the writings of three French architects explored these connections in greater detail, namely, Jousse (1642), Desargues (1643) and Derand (1643). In the 1730's, these connections were taken up anew by Gaurino (1737) and exhaustively by Fr‚zier (1737-1739) in his monumental three volume, Theory and practice of stonecutting. As the eighteenth century progressed, this subject emerged as the new field of stereotomy, which was subsequently subsumed as a category of descriptive geometry.
A class of books devoted specifically to architectural perspective emerged slowly. In the sixteenth century, there was a tradition of dedicating perspectival treatises to a number of professions (cf. p. ). This tradition continued in then nineteenth and twentieth centuries. A number of authors addressed their texts to painters and architects as, for example, Palaiseau (1818), Hummel (1825), Klette (1867), Longfellow (1901, 1908) and Denev (1948). There were numerous variants on this theme. Girardon (1850, 1900) addressed his work to schools of fine art, artists and architects; Holmes (1937, 1938, 1946, 1948, 1954, 1957, 1962, 1967) to artists, painters and art students; Clark (1936) to artists, architects and students, while Pyne (1870), specifically addressed young students and amateurs in architecture, painting. Mols (1816), extended the scope to painting, architecture, mechanics. Berger (1867) mentioned architects, construction workers, painters and amateurs. Adamo (1899) limited himself to architects and construction workers. Others included architects and designers, e.g. Sierp (1958, 1969,1972), White (1968, 1969,1974, 1976), Coulin (1966, 1962). Capelle (1969), addressed architects and engineers. Pizzigoni (1951), addressed painters, architects, scenographers and cinematographers, while Gratry (1855) included artists, painters, architects, engravers, decorators and all persons concerned with drawing.44
In the latter half of the nineteenth century, there evolved architectural treatises addressed specifically to architects, such as Schaap (1856), Schoen (1863), Krause (1876), Wright (1885, 1890, 1892, 1898) and Ferguson (1891, 1895). In rare cases, they were even more specialized as in Bailby's (1876) Complements of perspective. Application of linear perspective to the architectural decoration of ceilings. The first decade of the twentieth century saw no less than four new authors with works on perspective dedicated specifically to architects: Lawrence (1902, 1908, 1922, 1927, 1931, 1947), Ferguson (1903, 1915), Middleton (1903, 1907, 1915, 1919) and Hicks (1909). The next decades saw more examples with Rudd (1916), Dean (1933), Mashkov (1935), Bullen (1942), Retera (1946), Schutte (1949) and Woord (1953). The 1960's saw another upsurge, with Parrens (1962, 1967, 1973, 1971, 1982), Georghiu (1963), Schaarw„chter (1964, 1967), Gomolozewski (1966), Danielowski (1968, 1969, 1976, 1982) and Rosati (1969). More recent texts on architectural perspective, dedicated to architects, include Docci (1972), Suzin (1974) and Bonbon (1977).
In addition to this obvious level of books specifically devoted to architectural perspective, there were other connections between architecture and perspective which will become more apparent through a brief excursus on classification of architecture. In the mediaeval period written architectural knowledge was imparted mainly via the Vitruvian tradition and model books45 such as those of Villard de Honnecourt,46 which contained a variety of other topics including surveying, geometry, machines and drawing (fig. ). The fifteenth century saw a further distinction between books of mechanical inventions (e.g. Taccola)47 and those which focussed on architectural inventions (e.g. Francesco di Giorgio Martini), though vestiges of other topics remained in both. By the end of the fifteenth century, a distinction emerged between pleasurable drawings (e.g. Leonardo da Vinci, Windsor, Royal Collection) and useful drawings (e.g. Leonardo, Codice atlantico). In the sixteenth century there were further distinctions between useful mechanical drawings (e.g. Besson, cf. below p. ) and useful architectural drawings (e.g. Serlio), between modern (Serlio), and ancient buildings (Du Cerceau); between civil (Palladio) and military architecture (Cataneo).
In terms of perspective, there were works which dealt specifically with military architecture such as Specklin (1589, 1608) and Perret (1601, 1613). Perspectival fortifications gradually found their way into standard works on perspective such as Dubreuil (1642, etc.), Pozzo (1693, etc.), Bretez (1751); manuscripts such as Sovero (16__), Leturc (17__), as well as a considerable literature on military architecture proper by authors such as Marolois and Hondius, who also wrote treaties on perspective, Le Duc, Pagan, etc.
Meanwhile, (fig. ), there were also categories of literature to distinguish between sacred and secular architecture, or dealing with specific parts of buildings such as columns (Porta) or chimneys (Vredeman de Vries), or special problems such as gardening (Mollet, cf. below 2.3) or stonecutting, mentioned earlier.
Gradually more subtle distinctions emerged (fig. ). Drawings could deal with the past (buildings that had existed), the present (existing buildings), the future (planned buildings) or be a- temporal (imaginary buildings). Drawings of past buildings could be based on archaeological evidence (cf. fig. 84.1, 2, 4) or be artists' reconstructions (e.g. fig. 97.3) of what a building or monument might have looked like. Drawings of existing buildings could be idealized (e.g. fig. 86.1-4), realistic (e.g. fig. 85.2) or actually measured. A temporal drawings could involve possible buildings, such as Steven's anachronistic view of Emmaus (fig. 85.2), or purely fanciful constructions, as in some of Piranesi's more exotic prisons.
Perspective played a complex role in these developments. On the one hand, it undermined such distinctions by creating images which subjected archeological record (fig. 96.1), architectural plan (fig. 96.2), architectural phantasy (fig. 96.3), architectural reality (fig. 96.4) and artist's reconstruction (fig. 96.5) to the same spatial rules. On the other hand, precisely because of its window principle, which introduced a possibility of testing and measuring potential matches between representations and object, perspective was crucial in making these distinctions possible (cf. below 2.4).
The details of these distinctions cannot concern us here.48 It will suffice merely to note that their evolution went hand in hand with the rise of treatises on perspective and indeed a whole corpus of technical literature on drawing (fig. cf. below p. ). Already in the sixteenth century professions such as surveying developed their own literature49 and by the nineteenth century this had led to a special branch of plan drawing. Cartographers developed a literature on map drawing (cf. fig. ); engineers on engineering drawing and architects on architectural drawing. The distinction between existing and projected buildings led to a special branch of planned drawings called architectural drafting. A branch devoted specifically to shadows (considered earlier on p. ) also evolved which became linked in turn with geometry. At least some of these developments must be considered in more detail.
Hirschvogel (1544), had consciously cited perspective as bringing together architecture and geometry (cf. p. ). By the nineteenth century, these links between architecture and geometry had developed considerably. In some cases they involved traditional Euclidean geometry as in Toussaint's (1812) Simplified treatise of theoretical and practical geometry and architecture, Bennett's (1837) Original geometrical illustrations or the book of lines, circles, triangles, polygons indispensable to architects and Burns (1853) Illustrated London practical geometry and its application to architecture drawing.
Tabacchi (1844), was one of the first to link descriptive geometry, perspective and architecture. This Italian work was followed by a French treatise by Peyrat (1865): Descriptive geometry applied to graphical instructions in general, to architecture, drawing of perspective and the determination of shadows. The next decades brought Tessari (1860, 1883), Applications of descriptive geometry, The theory of shadows and of chiaroscuro...for the use of engineers, architects and designers. and Schreiber (1884 cf. 1822, etc.). Twentieth century examples included Smutz (1938), Fischer (1942) and Lippold (1949). In rare cases, such as Mondino's (1962), Perspective and the theory of shadows. Applications of descriptive geometry and central projection for the use of students of the faculty of architecture, perspective, descriptive geometry, shades and shadows and architecture were all discussed together. Meanwhile, each of these had also emerged as independent themes (cf. p. ).
In a general sense, architectural drawing had emerged with the study of Roman ruins in the fifteenth century (p. ). Even so, one of the first works specifically devoted to the subject was Nicholson's (1795), Principles of architecture containing the true method of drawing ichnography and orthography of objects, geometrical rules for shadows, which went through numerous editions (1809, 1827, 1831, 1841, 1848). Shortly afterward in France, Lagardette (1803), also a pioneer with respect to shades and shadows, published, New rules for the practice of drawing and wash drawing in architecture. As the nineteenth century progressed, there were efforts to relate architectural drawing within a larger context of technical drawing. For example, authors such as Weale (1841) and Burn (1854, 1856, 1860, 1882, 1893), wrote on architectural, mechanical and engineering drawing. Willson (1898) wrote on science, engineering and architectural drawing. Burg (1830) and Robinet (1842, 1855, 1859) wrote on architectural and mechanical drawing, a theme which Armengaud (1848) pursued in his Course of industrial drawing applied principally to mechanics and architecture. Henry des Vosges (1843, 1845, 1846), and Reid (1848, 1858, 1859), wrote on architecture and surveying. Pyne (1864, 1894) addressed operation builders and architects. Davidson (1869, 1871, 1882, 1896) addressed building constructors and architects. Linear drawing, as a branch which applied to all the building trades, was also emerging through authors such as Heissig (1855, 1863), Julien (1861) and Anonymous (1875).
Meanwhile, by the 1850's, a trend towards specialization was also evident, with books specifically on architectural drawing by Etex (1850), Graeb (1855), Gantz (1856), Spiers (1887, 1888, 1892, 1902, 1905), Edminster (1899, 1902). This trend continued in the twentieth century with Roberts (1906, 1907, 1916), Field (1922, 1932, 1943), Loundes (1930, 1935, 1938), Hake (1929, 1948) and Farey (1931, 1949). The nineteen sixties saw an upsurge in publications with Halse (1960), Bonfigli (1962), Coulin (1966), Mller (1967), Lockard (1968) and Jacoby (1969). More recently there were publications by Bruzda (1971), McGinty (1980) and K”nig (1979, 1984). Many of these were more primitive than their nineteenth century predecessors, mainly because traditional tasks in this field were increasingly coming within the realm of computers (see below p. ).
The nineteenth century also brought a gradual distinction between drawings of existing buildings (architectural drawing), and plans for further buildings (architectural drafting), though this varied from country to country. In France, for instance, both drawing and drafting, remained part of a more general concept of dessin. In England, there was again no standard term for drafting. Burn (1857) dealt with it under Ornamental drawing and architectural design, while Tuthill (1881, 1891, 1892, 1894, 1897, 1902, 1905, 1914, 1915) did so in, Practical lessons in architectural drawing or how to make working drawings and write the specifications of buildings. In Germany,,there evolved a branch of drawing concerned specifically with plans (Entwrfsdarstellungen) as in Seeger (1969). Drafting emerged as an independent field particularly in the United States, where Armengaud's (1848) Cours de dessin industriel, was translated as The practical draughtsman's book of industrial design forming a complete course of mechanical, engineering and architectural drawing (1853, etc., cf. Andr‚, 1874). In the twentieth century Hornung's (1910), Architectural drafting became a standard work with at least five subsequent editions (1949, 1954, 1955, 1966, 1971). Other significant works have included Stegman (1914, 1966), Fischer (1942), Barnes (1960) and Bellis (1961). But here, even more so than in the case of architectural drawing, the development of computer aided design (CAD) has been changing the field so rapidly that there is at present no standard textbook (see below p. ).
While there have been many innovations, there has also been a striking continuity of images and problems within architectural treatises. For instance, the vault has been a regular theme ever since Piero della Francesca used it in On perspective for painting. Serlio (1544, etc.) used it as did Lautensack (1564). In the seventeenth century, Dubreuil (1642-1649) used it a number of times and it remains a stock image as witnessed by Schaarw„chter's (1967) Perspectives for architecture. A similar continuity is evident in terms of problems. In the nineteenth century, authors such as Edwards (fig. 64.1-2, 1803) and Schreiber (1854) offered multiple perspectival views of a given building. Morgan (1950) provided a more systematic treatment of this problem in Architectural drawing, perspective, light and shadow, rendering, and the latest developments in computer aided design (p. ) can be seen as a direct elaboration of these principles. Or one might cite the example of Cloquet (1823), who offered both abstract geometrical and photograph-like perspectival drawingsw of various scenes in his Treatise of picturesque perspective, a method which was taken up by La Gournerie (1859, 1884, 1898) and which has become widespread in the last decades through a juxtaposition of photographs and line drawings (p. ). Authors such as Hiss (1985), who distinguish between three levels of perspective drawing--base perspective, design study and final perspective--have simply taken this approach one step further.
To understand the larger context of these developments it is necessary to return to the distinctions between pleasurable and useful drawings, mentioned earlier (p. ). In the sixteenth century useful drawings were divided into architectural and mechanical drawings. The latter category led to collections of different machines such as Besson and Ramelli and specific mechanical devices such as fountains and wells (e.g. Vredeman de Vries,1568). Such books had straightforward examples with little or no explanation. By the nineteenth century, books on specific mechanical devices such as mills (e.g. Böckler 16 ) with some explanation appeared. By the eighteenth century engineers such as Leupold (1727) moved in the direction of general principles of machines, an idea which F. Reuleaux developed in his Kinematics of machinery.50
To convey this knowledge also required developments in drawing techniques. Literature on drawing of machines, as noted earlier was frequently linked with architectural and engineering drawing, but gradually emerged as an independent branch of machine drawing or mechanical drawing.51 As in the case of architecture, further distinctions arose between existing and planned states, which led to specific books on machine drawing and machine drafting or design. Meanwhile, because all technical drawings involved constructions with the aid of instruments a distinction arose in Germany between construction drawing, and freehand drawing which applied to artistic and aesthetic domains. Construction drawing included applications to professions such as architecture and engineering and various trades such as bricklayers, carpenters, decorators, locksmiths, masons, etc. These distinctions varied from country to country. In France, dessin remained the general term for both technical and aesthetic domains. Industrial drawing became a term for technical applications involving professions, while linear drawing applied to trades in general. The Larousse Encyclopédie asserted that linear drawing included "tracing of working drawings, of elementary, descriptive and analytical geometry, ordinary and isometrical perspective, drawings of architecture, machines and topography."52 In England, construction drawing was frequently referred to as geometry applied to art (cf. fig. ), while freehand drawing was simply called drawing. In the United States these distinctions between useful and pleasurable drawing emerged as technical drawing versus fine arts.
If the precise relations between drawing and descriptive geometry or mathematics as a whole varied from country to country, the net result of these relations was the same everywhere. They introduced a vision of a single set of laws underlying the near infinite diversity of applications. As a result, in the nineteenth century, literature began to go in two opposing directions. One involved ever more branches of drawing: architectural, engineering, industrial, machine, map, plan, technical, etc. (fig. ). The other involved a quest for a universal graphic language which would encompass all these variants, which has been inherited by the realm of computer aided design and which helps explain why distinctions between technical and aesthetic drawing have never become as clear cut as some might have wished.
Even so there had been a number of developments specifically related to pleasurable or aesthetic drawings. At the beginning of the sixteenth century an implicit distinction had arisen between drawings of artifice (luxuries of the man made world) and nature (luxuries of the natural world. With respect to the former, Leonardo devoted a section of his Treatise on Painting to drapery,53 and drew various examples thereof in his notebooks. He also made several hundred drawings of grotesques, as well as numerous costumes, hats and ornaments. Some of these themes became subjects of separate books later in the century, as in the case of costumes (Amman )54 and ornament (Cock, Liefrinck, Dietterlin),55 while others such as hats, were treated in sections of drawing books (e.g. Vogtherr).56 By the nineteenth century most of these had become independent topics and in the case of ornament or costume there were even independent bibliographies.
As for nature, the situation was more complex. With respect to topics such as the human figure, a considerable literature evolved in the sixteenth century, proceeding in no less than four directions: anatomy (e.g. Leonardo, c. 1480-1815); proportion (Dürer, 1528); geometry (Stoer,1567) and subsequently, perspective (Lautensack, 1564; Cousin, le jeune, 1595, etc.). In the case of animals such as horses there were apparently treatises on their geometry, i.e. quadrature, and perspective by Foppa (148_) and Bramante (14__). In Soden Smith's catalogue of drawing books, the earliest examples date from the 1640's, namely, A. Cuyp's (1641), Various qaudruped animals drawn from life, and G. Fyt's (1642), Etchings of dogs.57
As far as aesthetic treatment was concerned, the botanical realms of nature evolved later. To be sure Leonardo, Dürer and their contemporaries drew a number of isolated examples of flowers, fruits and trees. Nonetheless, most sixteenth century treatises focussed on the useful and necessary aspects thereof, particularly with respect to their medicinal purposes. Indeed, in Soden Smith's list, the earliest aesthetic book on flowers was Syme's (1810), Practical directions for learning flowers, and in the case of trees, Laporte's Sketches of trees (1798-1801).58
Landscape, on the other hand, had begun in sixteenth century treatises on perspective such as the one edited by Rodler (1531), although the earliest examples recorded by Soden Smith were Stoer (1617) Preissler's (1734), Introduction which one can use in imitating beautiful landscapes in perspective, and Smith's (1797), Remarks on rural scenery with twenty etchings of cottages from nature.59 As for marine scenes, by the 1550's, Cock had produced a number of engravings of ships. In the seventeenth century, Robert Dudley produced remarkable manuscripts60 on the subject. Even so the earliest work in Soden Smith's list was an anonymous (1827) Book of shipping of various classes from the cutter to the first rate from drawings by Lieutenant Luna and others.61
Hence by the nineteenth century, what had begun as a simple category of pleasurable drawings, had spawned literature involving various aspects of the man made world (notably costumes, drapes, hats, grotesques, ornament) and nature. Soden Smith's standard catalogue, for instance, included 123 works on the human figure, 43 on animals, 7 on flowers, 29 on trees, 88 on landscapes and 13 on marine subjects.
5. Drawing Education
These practical changes in drawing were accompanied by institutional ones, which reflected both theoretical and philosophical developments, which had begun in the sixteenth century with efforts by Leonardo to create a systematic framework for representation. Lomazzo (1585), reduced these precepts to a list of five basic concepts: proportion, force (and motion), colour, light and perspective.62 De Piles63 and others produced their own variants on this list of ingredients. Bracquemond (1885) expanded it to include: drawing, colour, warmth, reflections (and chiaroscuro), value, outline (and modelling), line (mass and silhouette), perspective, effect, execution, ornament and decoration.64
As for the goals of art, each country seemed to go in a different direction. In Germany, the tradition of Dürer, Fürst, Testelin, Bloemart, Preissler and Herz emphasized geometrical drawing.65 In Italy, ever since Alberti, there had been an emphasis on composition and aspects of the story (istoria). In addition there were also major regional differences ranging from the Venetian preoccupation with colour, to the Florentine concern with chiaroscuro, to create effects of relief through line drawing. In the Netherlands, thinkers such as Van Mander and Hoogstraeten developed their own version of the story telling goal. Hoogstraeten, for instance, described nine types of story telling to correspond to each of the muses.66 In France, debates shifted to questions of technique, namely, to relative values of line versus colour, which sparked a major controversy between Poussinists and Rubenists.67 By the mid-seventeenth century it appeared that perspective was losing the significance it had once had. Indeed the decision of the French Academy to oust Bosse as professor of perspective (see above p. ) appeared only to confirm this trend. Then there is the as yet untold story of the role that perspective played in the eighteenth and nineteenth centuries as the academies of (fine) arts, particularly in Rome, Florence, Milan, Paris, London, Vienna, and Copenhagen became public institutions and, alas all too often, prisons for the imagination as members of the avante garde would one day complain.
At the same time there were other currents, which led in the opposite direction and were destined to make perspective a fundamental aspect of drawing. Part of this impetus came from philosophers such as Locke (1693)68 and Rousseau (1762),69 who decided that perspective should become a basic ingredient in the education of children.70 How to teach drawing properly, and what should be the goals of drawing became a matter of increasing debate. Leonardo da Vinci had considered at least three distinct goals of representation: at one extreme, nature; in between, models, and at the other extreme, imagination. By the latter seventeenth century, the ideas of Lairesse, which favoured the copying and imitation of models had gained ascendancy. By the later eighteenth century two intermediate positions found powerful new exponents. The Swiss philosopher of education, Pestalozzi (1746-1847) argued that models might be used to develop the imagination,71 while Thibaut (1757-1826), professor of perspective at the school of architecture in Paris argued that models and perspective might be used in nature drawing.
Pestalozzi's position was developed by Joseph Schmid (1809), who argued that this approach to imagination via models could be aided by the use of geometry and perspective which led, in turn to no less than five further responses in the early nineteenth century: 1) Reissmann (1801, etc.), who claimed that one could stimulate imagination via geometrical forms and perspective; 2) Boniface (1823), who emphasized perspective as a stimulus to the imagination; 3) Ramsauer (1821), who proposed the use of models, geometry, perspective and even some nature in stimulating imagination; 4) Francke (1833, 1836), who emphasized copying examples using perspective and; 5) Harnisch and Platz (1815), who favoured stigmographic drawing with some perspective. Later in the century there were two further responses; 6) Glinzer (1868), who insisted on dictation drawing to help the imagination, which method Pillet (1882) subsequently linked with perspective; 7) Soldan (1830) and Bes (1891, 1896), who believed that projection drawing and perspective offered a key to the imagination.
Meanwhile, Thibault's opposing school, which emphasized the use of models for nature drawing, led to two important responses. The chief of these was by his student, Thénot (1803-1857), who emphasized (1826) the use of perspective and some models. His work proved influential and was soon translated into Dutch (1829), German (1833, etc.), American (1834, etc.), English (1836, etc.) and subsequently Italian (1870). A second response to Thibaut came from Peter Schmid (1828) who emphasized models, but adcknowledged the use of perspective, shades and shadows in the process. This in turn inspired a clearer formulation by the brothers Ferdinand and Alexandre Dupuis (pl. 1820-1840), who argued for the use of both geometrical and perspectival models in the process of nature drawing.72 This approach won such acclaim that it was introduced into Germany in the next decades by Wolfgang (1825-1874), Fürstenberg (1854) and Domschke (1876), and into the Netherlands by Parv‚ (1852), Bergmann (1857) and Braet von Ueberfeldt (1863). The next generation saw a series of German books on nature drawing which emphasized the use of perspective, namely, Seeberger (1871), Huther (1872), Audel (1880), Kuchinka (1880), Lang (1880), Steigl (1880), Kajetan (1881) and Gennerich (1882).
The result of these developments was that perspective became integrated into the whole spectrum of drawing, ranging from natural objects, and man made models through purely imaginary objects as becomes evident from a brief examination of some of the key programmes. In the Encyclopédie (1754) drawing was defined as the art of imitating and the chief objects of study were listed as the human figure, animals, landscapes, draperies, flowers and fruits.73 Joseph Schmid's programme for imagination via models had five basic steps; 1) exercises for the education of the hand for drawing; 2) drawing exercises in making and inventing beautiful forms; 3) exercises which lead to development and strengthening of the imagination; 4) exercises in real or mathematical copying of natural objects and; 5) exercises in perspectival development.74 Peter Schmid's programme for nature drawing via models had four steps: 1) bodies of plane figures; 2) bodies of curved figures; 3) practical perspective; and 4) shades and shadows.75 His successors, the brothers Dupuis, took for granted the use of perspective in their programmes and concentrated on defining more precisely the models to be used. Ferdinand Dupuis outlined five levels of inorganic models: 1) simplest geometrical figures; 2) straight lined geometrical figures; 3) composite figures including characteristics of one and two; 4) stereometric figures; and 5) furniture, ceilings, pillars, columns and ornaments.76 Alexandre Dupuis had a four stage programme for more organic forms: 1) heads, plastercast, and live; 2) human form; 3) ornament; 4) flowers.77 Bergmann's programme (1857) read like a synthesis of the two: geometry, light, flowers, ornament, landscape, figures, animals, perspective, light and shade.78
The Universal dictionary of the nineteenth century ( 1870), in its article on drawing, cited a passage by Delacroix, which gave some indication of how important perspective had, in the meantime become:
Drawing, he said, is not to reproduce an object as it is, that being the task of the sculptor, but as it appears and this is the task of the person who draws and the painter. The latter achieves by means of gradations of tints that which the other began by means of proper disposition of lines. In a word it is perspective which one needs to place, not in the spirit, but in the eye of a student. I will say to the instructor: with your proportions and perspective by A plus B, you will not teach me other than truths and in art all is lies....Whatever be the object that a person who draws proposes to reproduce, they are still obliged above all to know and to respect the laws of perspective. There is linear perspective and aerial perspective. The former, for which descriptive geometry furnishes the rules, suffices for drawing, which only makes use of projections [and] contours. The second, which has as its object the apparent modifications, which plays of light and shade cause forms to undergo, finds its application in coloured images, either monochrome, as are drawings in sepin or china ink, or multicoloured. One might say that on a canvas, drawing gives the linear perspective and colour the aerial perspective.79
The latter part of this passage is particularly interesting because it suggests that the earleir debate of line versus colour, of the Rubenistes versus Poussinistes, had now been resolved through a synthesis of linear and colour perspective.
Even more instructive was the section on drawing in the Dictionary of Pedagogy (1881). The author emphasized the importance of perspective and noted that:the developed theory of perspective gave cognisance of the procedures imagined by geometers in order to obtain optical appearances in all cases in which it is necessary for a professional painter and especially an architect to know how to execute the same.80
It was conceded that: "the goal of study is drawing after nature but it is here that deeper study begins. In an elementary study one will not go beyond drawing based on a model."81 This elementary course was, however, considerably more complex there one might imagine as becomes clear from the teaching in elementary schools as reported in the Dictionary (1881):
Elementary Course
Tracing of straight lines and their division into equal parts. The evaluation of the relation of lines to one another; reproduction and evaluation of lines.
First principles of drawing of ornament. Circumferences regular polygons, stellated roses.
Copies of plasters representing ornaments in plan of low relief.
First notions of geometrical drawing and elements of perspective.
Geometrical representation in projection and perspectival representation in projection, then with shadows of geometrical solids and simple ordinary objects.
Geometrical drawing. Use (at the board) of instruments serving the tracing of straight lines and circumferences: rule, compass, square and protractor.
To limit oneself in this part of the course, to have pupils understand the use of these instruments of which they will acquire skilled use in the upper level course.
Upper Level Course
Drawing with a raised hand
Drawing after an engraving and after a relief of purely geometrical ornaments, mouldings, ovals, heart-shaped spokes, indentures, etc.
Drawing after engraving and relief ornaments taking their form from the realm of vegetation, leaves, flowers, fruits, little palmettes, foliated scrolls, etc.
Elementary notions on the architectural orders given on the blackgoard by the teacher (3 lessons).
Drawing of the human body: its parts and proportions.
Geometrical drawing. Execution on paper, with the help of instruments, geometrical traces which have been made on the board in the middle level course.
Principles of wash drawing with flat tints.
Drawing reproducing motifs of decoration of plane surfaces or of a low relief: tiling, parquetry, windows, panels, ceilings. Wash drawing in china ink and in the colour of some of these drawings.
To bring into relief with dimension figures, with sides, and geometrical representation in projection of geometrical solids and simple objects such as assemblages of timber, woodwork, exterior dispositions of apparatus of stone, big pieces of ironwork, furniture of the most ordinary type, etc. Use of wash drawing to express the nature of materials. Wash drawings of plans and maps.82
Most interesting of all was the philosophy that lay behind this process. It was not just a matter of teaching the gifted to draw well. There was a conviction that the study of:
drawing should not only guide a group of these who would devote themselves to acquire the talent of representing visible things either by pure imitation or in imagining and in inventing in this way to the point of art, but also for those who would not succeed in acquiring this talent or who would only acquire it in a feeble measure, this study, if one founded it on the imitation of excellent models would teach them what is, in fact, exact or inexact, correct or incorrect, beautiful or ugly, graceful or ungraceful, suitable and unsuitable, such that it would thus teach to see better and to judge better and that it would form, finally, [a good] eye and taste, the usefulness of which is almost universal.83
This passage helps to understand why drawing had become universal, why methods which had in the sixteenth century been aimed only at a select body of professions (painters, architects, goldsmiths, etc.), had spread in the seventeenth and eighteenth centuries to artistic schools and academies, and in the nineteenth, to trades, secondary schools, primary schools and ultimately all interested laymen. Drawing, and the perspective it entailed had become part of learning how to see, a basic ingredient to a level of taste which was seen as equivalent with civilization itself. Ironically once perspective had become so integral a part of drawing, it became typical to think of it as a part of drawing and hence overlook the many titles relating to perspective as they appeared.
The changing relations between mathematics and perspective are no less complex. Any attempt at a thorough treatment would lead far more deeply into the history of mathematics than is here possible. For our purposes it must suffice if we outline a few major shifts in the classification of mathematics and mention their consequences for books on perspective.
The latter fifteenth century brought a particular fascination with the concept of proportion which led the Dominican friar, Luca Pacioli (1494), to compose his monumental Compendium of geometry, proportion and proportionality. For Pacioli, perspective was a branch of proportion and for this reason his compendium became the first printed book to deal with perspective In a subsequent sermon in August 1508 (see p. ) he praised the universal applicability of proportion, mentioning perspective in connection with painting in a long list of disciplines which it affected, namely, theology, philosophy, medicine, astronomy, chorography and cosmography, architecture, inventors of machines, painters, sculptors, musicians, poets, rhetoricians, grammarians, lawyers, mathematicians and the pious. In the latter part of the century Belli (1573), wrote a book on measurement of distance, heights and depths to which he added three books on proportion and proportionality. This fascination with proportion as an organizing category was an inspiration for the proportional compass or sector in the early seventeenth century and the resulting books by Faulhaber, Bramer, etc. maintained the idea of perspective as a branch of proportion (see below 2.1).
Meanwhile, Euclid had included surveying propositions in his Optics, and during the mediaeval Arabic tradition, surveying emerged as an important part of optics. With the development of perspective there were also connections between perspective and surveying, whereby perspective became associated with measurement in the tradition of Drer, and actually identified with measurement by the author of the book edited by Rodler (1531). But since Boethius it had become customary to treat the etymology of geometry literally, to mean measurement of the earth, (yn metrwv). As a result geometry and perspective were in some senses synonymous.
However, the systematic programme of reviving classical mathematics which began at Urbino with Commandino and Guidobaldo del Monte, meant that perspective was increasingly listed under optics as a subalternate branch of geometry. This process continued after Paris had replaced Urbino as the European centre for mathematical studies (cf. p. ). Indeed, by the sixteen thirties, thinkers such as Descartes, Pascal and Desargues had raised the level of abstraction to a point which implicitly required distinctions between different levels. In addition to their high level, there emerged a level of high divulgation (e.g. Bosse), popularization (e.g. Dubreuil), and several levels of simplification (e.g. Frobes), which echoed in readers digest form the ideas of others.
The level of high mathematics was continued in the eighteenth century by thinkers such as Brooke Taylor, Lambert and Euler. In the final decades of the century this tradition led individuals such as Tinseau d'Amondas (1777, 1780) and Monge (1785, 1795, 1798, 1799) to integrate developments in practical disciplines such as architecture, dialling, stonecutting, (Fr‚zier), perspective, and surveying in the form of descriptive geometry.
In the first decades of the nineteenth century this new field evolved mainly in Paris, through Monge (1811, etc.), Hachette (1815, 1817, 1820, 1827, 1838), Potier (1816, 1817) and Vallee (1819, 1822, 1828), Duchesne (1828, 1829), Olivier (1831, 1839, 1840, 1842, 1844, 1844, 1847), Leroy (1834, 1837, 1838, 1842, 1845, 1846), Bardin (1837), Mathieu (1843), Cirodde (1844), Narrien (1846), Biston (1848), Poudra (1849). It soon spread to other centres including Milan (Tabacchi, 1813), Karlsruhe (Schreiber, 1822, 1828, 1833, 1839), Rome (Sereni 1826), New York (Davies 1826, 1844), Cambridge (Hayward 1829), Berlin (Wolff 1835, 1847), Munich (Haindl 1835), Nrnberg (Gugler 1842), Wintherthur (Ziegler 1843), Vienna (Stampfl 1847), Pavia (Pasi 1844) and Padua (Tabacchi 1844).
Descriptive geometry now became a general heading under which perspective was classed. Tabacchi (1813) was one of the first to relate precepts of descriptive geometry to drawing. Cloquet (1823) consciously became one of the first to include descriptive geometry in his New elementary treatise of perspective. As he explained in his preface:
It happened to me on several occasions that I asked artists who told of their desire to know perspective, whether they had any hint of the first elements of geometry. They replied negatively saying that they had no need thereof, that they wished to know only one perspective or rather painter's perspective. And yet there is only one sole perspective and in my opinion expressing such a desire would be like asking to write without wanting to learn to read. My way of reasoning soon convinced them, but never quite persuaded them altogether. I did, however, observe that among those who turned to me, those who resigned themselves to study the preliminary principles, learned perspective easily and truly, a science of which the painter should not be ignorant.This has led me to reduce the elements of perspective to its essential elements and place them at the disposition of readers without any notion of mathematics, which is fortunately quite rare. I have therefore divided this work into five parts. The first part deals with elementary geometry which I have stripped of all matters foreign to our subject, such as those concerning the measurement of surfaces, solids, etc. The second contains purely elementary principles of descriptive geometry, which is nothing more than a consequence or application of the the principles of part one. The third deals with that part of optics which concerns our subject directly, namely optics considered with respect to painting rather than to physics. The fourth deals with the rules of projections of shadows very necessary not only for painters, but also for architects, designers, etc. Finally the fifth deals with perspective which is the last and principal subject with which I propose to deal.84
Haindl (1835), included descriptive geometry as part one of his, Course on the science of drawing and, as mentioned earlier, Brisson (1827) developed a Theory of shadows and perspective, which was subsequently appended to editions of Monge: (e.g. 1838, 1847, etc.). In the second half of the nineteenth century Paris remained a major centre for descriptive geometry with a number of further editions of Leroy (1850, 1859, 1861, 1865, 1867) and Olivier (1851, 1863, 1866), a popular new work by Catalan (1850, 1857, 1861, 1864, 1867, 1868) and others by Babinet (1850), Amiot (1852, 1869), Bisson (1856), La Gournerie (1860), Hughes (1864), Du Peyrat (1865), Dufailly (1869, 1894), Pillet (1879, 1899), Lagrange (1877) and Martin (1897).
Cunningham (1867) in his, Notes on the historical method and technical importance of descriptive geometry,85 assessed these developments in terms of three trends: a move towards descriptive geometry as a science in France; in relation to the art of drawing in Germany and an independent movement in England, which emphasized practical geometry (fig. ). With the advantage of hindsight it is possible to recognize further trends which offer a corrective to this picture. For instance, in the 1850's, London also became an important publishing centre for texts on descriptive geometry by Wooley (1850), Heather (1851), who was translating the ideas of Monge; Binns (1865, 1867), and Millar (1878). In New York, interest which had been sparked by Davies (1826, 1844, etc.) was developed by Warren (1860, etc.) and Mahan (1867). In Italy, interest in descriptive geometry was focussed in Northern cities such as Padua (Bellavitis, 1851), Turin (Tessari 1880, 1883), and Milan (Aschieri, 1883; Suini, 1886).
Meanwhile, there were four German speaking centres which emerged: Leipzig (e.g. Bnau, 1852; Schreiber, 1865 and Wiener 1885), Berlin (Stoevesandt, 1856; Pohlke, 1860, 1873; Wolff, 1861; Flohr, 1866); Vienna (Heissig, 1859; Schlesinger 1871; Fialkowski 1882) and Stuttgart, (Gugler, 1856, 1875, 1880; B”klen, 1866; Riess, 1871; Vonderlinn, 1888, 1893). Links between publishing houses meant that works appeared in both Leipzig and Berlin (e.g. Schmidt, 1869; Mller 1872; Papperitz, 1893) or in both Leipzig and Vienna (Peschka, 1883). A network of mathematical centres thus emergd which were simultaneously at the forefront of other developments also.
Higher or Projective Geometry
Ancient mathematicians had distinguished clearly between arithmetic and geometry in terms of number versus extension. The mediaeval and Renaissance emphasis on practical geometry which equated geometry and measurement undermined the clarity of this distinction. In the 1630's Desargues' work suggested a more abstract approach to geometry as a study of projections. Even so almost two centuries passed before Victor Poncelet, left behind by Napoleon, imprisoned on the plains of Saratov (1813-1914), wrote his fundamental book on projective geometry which appeared in 1822. Durell (1838) wrote an early English work on the subject. K. G. Ch. von Staudt (1847), in his Geometry of position established a basic distinction between descriptive and metrical geometry. The next decades saw further work by Hertzer (1865), Poncelet (1865), and Cremona (1871, 1878, 1885, 1893). Projective geometry, which also became known as higher geometry, now emerged as a heading under which both descriptive geometry and perspective were classed.
Meanwhile, the frontiers of projective geometry were being expanded by new discoveries. Traditionally it had been assumed that Euclid provided the foundation for the whole of geometry. Bolyai (1832), Lobachevsky (1840) and Riemann (1854) demonstrated that non-Euclidean geometries were possible. French translations of Lobachevsky (1866), Bolyai (1868) and Riemann (1870) by Houel made these ideas accessible and forced thinkers to reconsider the nature of projective space at the very time that the optical researches of Hering and Helmholtz were challenging thinkers to reconsider the nature of visual space.
By the 1880's thinkers were writing on projective and descriptive geometry together: e.g. Aschieri (1883, cf. 1895), Peschka (1883, 1889), Vries de Hecklingen (1902), Gallucci (1935), Mondino (1961) and Nannoni (1978). Doehlemann (1898, 1901, 1905, 1918, 1924) wrote books on both projective geometry and perspective (1916, 1919, 1928). Schaufler (1906) wrote an Introduction to perspective and projective geometry and more recently Morehead (1955) wrote. Perspective and projective geometries: a comparison. Perspective is still frequently classed as a branch of projective geometry. Other developments have, however, shifted this classification anew.
Analytic and Algebraic Geometry
Serious interest in conic sections began in the sixteenth century with Werner (1522) and with Commandino's (1566) edition of Apollonius and Serenus. In the 1630's, Descartes' explorations of analytical geometry (cf. above p. ) drew attention to different kinds of lines such as ellipses, parabolas and hyperbolas which, it was recognized were all sections of cones. Some of these properties were explored by Descartes' colleagues, Mydorge (1631, 1639) and Pascal (1640). The following decades saw important works on conic sections by Schooten (1646, 1656), Saint Vincent (1847), Courcier (1662) and La Hire (1673). These interests continued in the eighteenth century with thinkers such as Le Poivre (1704), L'Hospital (1707, etc.) and Simson (1735, etc.) and it was gradually recognized that conic sections were connected with more universal principles. Murdoch (1746), for instance, wrote Genesis of Newton's curves by shadows or elements of universal perspective illustrated with examples of conic sections and lines of third order. and Vince (1800), Elements of conic sections as preparatory to reading of Sir Isaac Newton's Principia.
The 1820's, which saw the emergence of projective geometry, also saw new attention given to conic sections. Plcker (1826-1827), for instance, explored new methods of visualizing these problems and Ohm (1826) wrote a basic work on Analytical and higher geometry in its elements with particular attention to the theory of conic sections. The 1850's and 1860's saw much new activity in this field, including Salomon (1851, 1870), Puckle's (1854), Elementary treatise on conic sections and algebraic geometry, Gerling (1865, 1875) and Staudigl (1875).
Leipzig became a centre for these interests. M”bius (1855), for example, introduced homogeneous coordinates in projective space. The work of Salmon was translated from the English and appeared in a number of editions (1860, 1870, 1873, 1878, 1887, 1888, 1898). Erler (1862, 1893) wrote on analytic geometry and conic sections. There were numerous works on conic sections: Geiser (1866, 1867), Eckhardt (1866), Drach (1867), Steiner (1867), Schr”ter (1867), and Grelle (1869). These interests continued into the twentieth century with Leipzig remaining as a centre until the first world war: e.g. Dette (1909) and Staude (1910).
By this time thinkers were exploring connections with other branches of geometry. For example, Rudolphi (1914, cf. 1912) published Analytical geometry of space in relation to descriptive geometry. In Italy, there was a particular interest in comparing analytic and projective geometry with authors such as Martinetti (1926), Fano (1930, 1957), Campedelli (1945, 1970) and Chisini (1960). The work of Van der Waerden (1931-1939) established algebraic geometry as a more universal class which subsumed projective geometry, descriptive geometry and perspective. Van der Waerden limited himself to exploring algebraic varieties of projective space. Weil (1962) in another fundamental contribution considered algebraic varieties of affine space. As a result of these developments, perspective, our chief means of visualizing the world tends to be classed under algebraic geometry which is non-visual.
Two other basic developments in mathematics deserve brief mention. Felix Klein's (1872) Erlangen program proposed that geometries should be classed according to groups of transformations that can be applied without changing basic concepts, axioms and theorems.86 This led Wolf and Wolff (1956)87 to search the secrets of geometry in nature in terms of thirteen transformations (fig. ) while thinkers such as Coxeter and Greitzer (1967)88 reduced these to seven basic types in their genealogy of transformations (fig. ). Efforts to class geometry have also varied considerably. Coxeter (1969)89 distinguished simply between absolute and affine geometry. Psychologists such as Hagen (1986)90 distinguished four kinds of geometry: metric, similarity, affine and projective. Architects such as Steadman and March (1972),91 drawing on analogies between geometry and geography, whereby methods of geometry become different kinds of mapping, identified six basic types: identity, isometry, similarity, affinity, perspectivity and topology (fig. ). In the process perspective has effectively lost any privileged status as a means of spatial organization.
Perspective is often described as something particularly linked with the Renaissance which continued until the nineteenth century and then died out. In the opening chapters we showed that this was not the case, that perspective has in fact continued unabated to the present. In this chapter we explored an underlying cause for its apparent demise. Perspective never became an independent concept and was therefore subject the the vagaries of classification of four branches of learning: optics, architecture, drawing and geometry. This had two important consequences. First, changes in classification, particularly in geometry, have shifted the definition of what actually constitutes perspective. In the nineteenth century, perspective included not only one, two, and three point perspective but also various branches of parallel perspective. In the twentieth century, the scope of the term has frequently been restricted to one, two and three point perspective, whereas all parallel versions are classed as projections (e.g. fig. ).92 Indeed, some would see perspective merely as an example of dilatation, as one of a number of mathematical transformations with no special role in spatial organization. A second consequence has been no less dramatic. A good deal of literature pertaining to perspective has simply been classed as part of the larger fields. Hence, although work on perspective has continued, it has been classed as work in optics, architecture, drawing or geometry.
Our bibliography has ignored the artificial barriers imposed by these fields, collecting hitherto disparate materials relating to perspective. Part one has created a framework for a larger picture of the phenomenon of perspective. Part two will explore the consequences of this phenomenon. 98 97
Last Update: August 4, 1998