The Perspective Construction of Masaccio's 'Trinity ... · The Perspective Construction of Masaccio's Trinity Fresco and Medieval Astronomical Graphics ... defined coordinates of

The Perspective Construction of Masaccio's "Trinity" Fresco and Medieval AstronomicalGraphicsAuthor(s): Jane Andrews AikenSource: Artibus et Historiae, Vol. 16, No. 31 (1995), pp. 171-187Published by: IRSA s.c.Stable URL: http://www.jstor.org/stable/1483503Accessed: 11/11/2010 17:41

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JANE ANDREWS AIKEN

The Perspective Construction of Masaccio's Trinity Fresco and Medieval Astronomical Graphics

For James S. Ackerman

Quite beyond the solemn reality brought to bear on the central mystery of the Christian faith, Masaccio's Holy Trinity fresco has played a pivotal role in the history of art as both a definitive example of early Renaissance linear perspective and as a kind of prophetic forerunner of the perspective method discussed nearly a decade later by Leon Battista Alberti.' While Alberti's Della Pittura of 1435 may be the first written document to articulate a new humanist ideal of painting in which visual appearances are controlled by geometric principles embedded in nature, Masaccio's fresco of the Trinity is the first extant painting fully informed by that ideal.2 The magnificent vault arching over the austere figures in Masaccio's fresco of the mid-1420s [Fig. 1] is an utterly convincing illusion of architectural form extending into space, and for centuries it has been justly celebrated on that account. Yet it is not an historical ex- emplar devoid of uncertainties. It has been difficult, for instance, to determine exactly where the figures of the Virgin and St. John stand with respect to the projected ground plane, and ascertaining the position of God the Father's feet has proved a particularly mystifying problem. So far as the perspective construction itself is concerned, there is little agreement among scholars about either the vertical position of the projected cen-

tric point on the wall or the distance of the viewer from the painting.3

These are important issues for art historians to resolve, and much time has been spent in considering them within the context of the Renaissance awakening. Yet having succumbed to the fascinating pursuit of Albertian consistencies, are we really any closer to understanding the method and conceptual frame- work of Masaccio's perspective scheme except in the limiting terms proposed by Alberti? Was Masaccio simply unable to master the difficult di sotto in su projection that would have suc- cessfully foreshortened the awe inspiring figures in the foreground of his fresco? Is it because the fresco has been moved several times that we are unable to decode Masaccio's perspective projection?4 Or is the stringently rational context of Alberti's Della Pittura really appropriate for understanding a painting created in Florence during the 1420s, when one might still expect to find a fluid dialogue between reason and faith in an image of the Corpus Domini executed for the conservative Dominican church of Santa Maria Novella?

Certainly Joseph Polzer's detailed photographs of the points and lines embedded in the fresh plaster provide strong evidence that the illusionistic impact of the vault depended on many dif-

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172

THE PERSPECTIVE CONSTRUCTION OF MASACCIO'S TRINITY FRESCO

ferent artistic techniques.5 It needs now to be said that this diversity of techniques includes orthographic, conical, and stereographic methods made familiar to late Medieval painters through instructive working drawings of architects and instrument makers as much as through practical geometry texts that had gov- erned the artist's early education.6 If the very complexity of the Trinity fresco vault projection has long encouraged art historians to conjecture Filippo Brunelleschi's involvement in its plan- ning, the diversity of the projection techniques discovered there would seem all the more to confirm the architect's participation in this complex project.7 Perhaps more importantly, as Polzer has shown, that diversity of means is compellingly unified both pictorially and theoretically at the level of mathematics and measurement. The imaginative sweep of Masaccio's accomplish- ment is not to be found solely in the precise ordering of lines and planes, however, for he (or more likely Brunelleschi) dis- carded earlier and more tentative experiments in favor of a rationally consistent method of structuring his own arching sepulchral vault which drew on and mirrored the mathematically defined coordinates of the vault of the heavens.

It is the aim of the following essay to show that the one pre- existing graphic tradition of great authority for projecting these mathematically regulated and symbolically charged spatial coordinates was the tradition of medieval astronomical diagrams. This tradition was not only useful in practical detail, but it was also intrinsically suggestive to early perspectivists, and prob- ably determinative with respect to the special viewing circum- stances presented by the Trinity. Not only did this graphic tradition take into account the position of the viewer looking intently upward; its most familiar projections were ordered according to the exemplary symmetries of a divinely created cosmos. The orthographic and stereographic projections of medieval astronomers and the common ground they shared with mathematical diagrams provided a readily available source to Masaccio and Brunelleschi of a full range of necessary diagramming techniques at the same time that they affirmed the mathematical order believed to control all of nature. To draw on such a tradition was not only an act of great practical consequence for painters; it was an affirmation of great conceptual force.

Masaccio planned the entire structure of the Trinity fresco in a most deliberate and mathematical way. As Joseph Polzer has shown, he controlled the composition through the rational forces of measurement and geometry by initially dividing the principal pictorial field into three squares averaging approximately 211 cm wide and 207 cm high, the skewing from perfect squares being accounted for by the moving of the fresco or by the difficulty of maintaining constant pressure on ropes when snapping lines or describing arcs over a considerable distance [Fig. 2].8 The bottom two squares are made to come out right within the

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rectangular field of the composition by overlapping them from the base of the ledge on which the donors kneel to the edge of the painted chapel floor. Finally, the top square is inscribed with the illusionistically receding semi-circular ribs of the barrel vault.

What distinguished Masaccio's squares and circles from those commonly used by medieval predecessors and set painting on a new path was his insistence that the surface divisions be linked to the projection establishing the apparent recession of a barrel vault [Fig. 3].9 As Polzer has pointed out, the bond between surface geometry and the illusion of recession was ac- complished by having the base line of the top square delimit the springing of the barrel vault in the distance and by further subdivision of the top square at its midline, marking the springing of the barrel vault in the extreme foreground.10 From a further study of the vault, it becomes clear that Masaccio, despite the perceived regularity of the apparent recession, deliberately adjusted the standard (that is, Albertian) components of a Renaissance perspective construction to achieve maximum visual appeal as well as to assert the power of the surface grid."1 The eye level and the horizon line, for instance, coincide with the base of the middle square, and the "viewer's eye" (or centric point of the projection) is located at the base of the central axis of the composition. Moreover, Masaccio insisted on the formal control of the basic square module by calculating a "viewing distance" equal to the length of the side of a square. Thus the "viewer" in Masaccio's perspective scheme was fully integrated into the overall surface geometry of the composition, and the hypotheti- cal person stationed in front of the painting with one eye closed

was, in fact, reduced to a potent mathematical entity.

While Professor Polzer has shown that the precise confluence of circle, square, and projection of the vault in

depth restricts the perspective projection to the basic square module, I hope to show that the artist's surface geometry was joined in both a practical and symbolically provocative way to a rigorous, mathematical interpretation of how points, lines, and planes behave in the ideal worlds of Euclidean geometry and medieval mathematical astronomy. The development of Renais- sance perspective will be seen to be dependent as much on knowledge of a long mathematical past and on placing under heavy subscription the ancient and medieval tradition of mathematical graphics as on any recently enhanced acuity.

Masaccio's combination of squares, double squares, circles, and semi-circles frankly invokes the traditional values ascribed to these presumably perfect forms in medieval as well as Re- naissance aesthetics and theology. As Rona Goffen has argued, "it seems likely that Masaccio's architecture is intended as a mathematical expression of God's perfection and harmony, wor- thy of the 'real tabernacle' of the Lord..." 12 Perhaps more appro-

174


priate for the growing secular tastes of the Renaissance, this overlapping combination of squares and circles also alludes to the almost irresistible Vitruvian symbol of a man contained in the circle and the square and thus to the assumed affinity between microcosm and macrocosm.13 The rhetorical intent of Masaccio's geometry is further suggested by the location of the incised centric point, denoting the height of the "viewer," at almost 3 braccia above the church floor, a measure which carries with it the same connotations of the ideal as the circles and squares of the surface geometry.14 While Alberti would main- tain that this was the height of the average viewer, giving it a seemingly practical sanction, it fits neatly into the Vitruvian scheme of ideal human proportions. In addition, it is a commonly known symbolic height in late medieval guide books to Jerusalem, where 3braccia is proclaimed to be the height of the perfect man, Christ.15 Despite the fact that the location of the centric point coincides with a reasonable viewing height, its placement confirms Masaccio's attention to non-physical and non- visual considerations associated with the time honored symbolic power of numbers as well as the purity of mathematical relationships and analogues. That one should find such doubly potent symbols of perfection in an image of the Trinity in Santa Maria Novella is in keeping with the central role played by the Corpus Domini in the sacerdotal life of this conservative Dominican Church.16 If Masaccio's fresco was adventurous, even radical, in its aggressive imitation of a powerful, physically present nature, the intellectual context of his grid and projection systems, as well as the newly rationalized aesthetic on which they depended, remained firmly linked to a traditional and highly suggestive religious interpretation of natural order in which mathematics func- tions as a bridge between concrete, sensible reality and univer- sal or divine truth.17

Quite beyond shape and measure conveying meaning in an obvious and frankly didactic way, the points, lines, and planes which make sense to many as surface geometry, medieval mathematicians would have understood within the broader context of a mathematical graphics tradition intent on explaining another kind of absolute perfection, the continually changing relationships among the coordinate systems of a vast and earth-cen- tered universe as those systems were projected onto a plane surface. These projections were a part of an unbroken tradition of mathematical diagramming techniques dating back at least to the 4th century B.C. The many different diagrams bound by this tradition were found in widely circulated copies of ancient texts by Euclid, Archimedes, and Ptolemy, medieval commentaries by Messahala, Jordanus de Nemore, and Campanus of Novara, as well as in the practical geometry tracts that formed the foun- dation of an artist's education in the Florentineabbaco schools.'1 The precision with which some of the still visible construction

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lines were scratched into the wet surface of the Trinity offers some proof that Masaccio (or more likely Brunelleschi) was not only familiar with this graphic tradition but even painstakingly followed its rules.19

Not merely the result of convenience, the compositional grid of the Trinity is the product of a highly conflated application of different lines of mathematical reasoning to a spatial problem whose main features derive directly from the astronomical conventions of the day. First of all, Masaccio's apparent use of a centric point to designate the projection of a ray (in this case, the principal line of sight) onto the plane of projection, together with the right angle relationship of ray to plane were not only defined by Alberti in 1435 but were typical aspects of medieval astronomical projections.20 Also, certain lines, generally regarded as mere surface marks by art historians, would have been inter- preted by mathematicians and anyone familiar with the astrolabe [Fig. 4], the most popular astronomical siting device of the

175

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5) Diagram of the projection of the (cTrinity)) fresco vault ribs according to J.Polzer. Redrawn by Gorman.

176


late Middle Ages, as projections of planes perpendicular to the plane of representation.

Called trace lines by present day mathematicians, there should be little mystery surrounding Masaccio's use of these lines, for they appear commonly in ancient and medieval diagrams of cones and pyramids. In astronomical projections, trace lines indicate critical celestial relationships, the most important for the artist being the projection of the horizon plane as a trace line.21 Accordingly, in Masaccio's fresco the base line of the middle square forms not only a segment of the surface grid but also a trace line designating the horizon plane rotated 900 with respect to the fresco surface.22 In addition, when working out the proportionate diminution of the spaces between the ribs of the barrel vault, Masaccio can be shown to have regarded the central vertical axis of the plane of representation as a traceline of the plane of projection (or picture plane) rotated 900 relative to the physical surface of the fresco.23 Finally, to think of the plane of projection as coincident with, but separate from, the plane of representation is another familiar and fundamental characteristic of ancient and medieval mathematical graphics.24

Aside from depending on geometrically obvious relationships (such as right angles) and ancient graphic protocols (like trace lines) to effect his projection in depth, Masaccio developed the rate at which the ribs of the barrel vault appear to diminish according to standards and practices familiar to his contemporaries from the stereographic projections found on the astrolabe, these more elaborate projections also being dependent on the simple graphic formulations described above.25 Although in 1435 Alberti claimed to have "invented" a newly rigorous method of perspective projection, in the 1420s Masaccio had already developed a mathematically consistent way to project apparent diminution and, like Alberti, had arrived at an understanding of the function of the so-called distance or lateral vanishing point construction from the diagrammatic techniques of mathematical astronomy.26

The several interrelated projection systems of the astrolabe and the literature describing both the theory and construction of this astronomer's siting instrument could be of special interest to the art historian for many reasons, the most verbally suggestive of which is that texts on astrolabic projection assigned to the projection point "the capacity to see," and characterized it as an "observing point," thus implying a consonance between ab- straction and actual viewing that would have appealed to the artist interested in quantifiable realities.27 Equally interesting for the art historian, two of the projections on the surface of the astrolabe appear to be foreshortened illusions of three-dimen- sionally extended spatial coordinates. While they are not actu- ally foreshortened, all the astrolabic projection systems together produce a visually cohesive, triumphantly coherent, and formally

consistent analysis of geometrically controlled spatial relationships.28

Masaccio could not have been alone in his admiration of this geometricum instrumentum, generally acknowledged by medieval astronomers to be the mirror of a geometrically perfect universe, since the astrolabe became the habitual attribute of Painting as symbolically personified by other Florentine artists seeking to represent rational order.29 Long before such Re- naissance representations, however, the astrolabe had become a symbol of the divinely ordered universe, as the seraphic be- ings holding astrolabes and encircling the archivolts of the Royal Portal of Chartres attest.30 Indeed, no less a figure than Boccaccio may have been thinking of this fertile conjunction of painting, mathematical predictability, and the heavens when he characterized Giotto as "so extraordinary a genius, that there was noth- ing Nature, the mother of all things, displays to us by the eternal revolutions of the heavens, that he could not recreate with pen- cil, pen, or brush so faithfully, that it hardly seemed a copy, but rather the thing itself."31

More technically, and beyond the felicities of symbol and metaphor, the projection of the semi-circular ribs of the Trinity barrel vault is directly analogous to the astrolabist's method of projecting almucantars (circles of celestial latitude) ranging above the observed horizon. In both cases, circular and equally spaced coordinate divisions of a spherical surface are projected onto a plane surface, and in each case a precisely located observer is involved. Finally (and most characteristic of stereographic projection), in both fresco and astrolabe, the construction determines the relative position of projected centers and radii of the circular divisions along a line ultimately falling on the central axis of the plane of representation.32

According to Professor Polzer's hypothesis, Masaccio determined the ribs of the barrel vault by extending lines from a single point (coincident with the horizon line and marking the distance of the viewer as equivalent to one side of the grid square) to nine points incised on the left radius of the foremost vault rib [Fig. 5].33 Where these extended lines cut the inscribed central vertical axis, Masaccio located the centers of the rib semi-circles.34 The radius of each rib arc was then extended from its center on the vertical axis to the orthogonal marking the left lateral edge of the vault so that the angle incidence of the pair of orthogonals delimiting the vault width (and ultimately the position of the centric point) determined the decreasing length of the diameter of each arc. Radius 1, then, was used to generate the front face of the foremost arc and radius 9 the back side of the most distant arc, with radii 2-8 forming the base of intervening arcs. With this method of construction Masaccio controlled the rate at which the curved vault ribs appear to diminish into the distance in terms directly related to the surface geometry and to the methods of

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6) Computer analysis of the projection of the ((Trinity)) fresco vault ribs using J. Polzer's method of projection and the measurements of J. V. Field, R. Lunardi, and T. B. Settle. Com- puted and drawn by T. Slater.

7) Stereographic projection of almucantars or celestial latitudes onto the face of an astrolabe. Photo: George V. Kelvin Science Graphics. Redrawn by Gorman.

astrolabic projection. Masaccio also produced an illusion of depth without any direct reference to a projected floor plane, a fact that surely accounts for the difficulties encountered in determining the exact location of the holy figures surrounding Christ.35

Any one who attempts to analyze what for Masaccio were the measured certainties of the Trinity is quickly brought to an appreciation of the difficulty of the task by being confronted with disagreement among scholars as to the actual measurements and their meaning. Because the recently executed measurements taken by J. V. Field, R. Lunardi, and T. B. Settle do not

confirm all of Professor Polzer's dimensions, it has seemed pru- dent to rework the latter's proposed construction method using the numbers of Field and her colleagues.36 The resulting diagram [Fig. 6], showing distances between the vault ribs (Al, viii, vii...i) and the distance of the eye to the central axis or intersection (D1 to D3), was generated by computer.37 While absolute agreement is not obtained between the hand generated and computer generated measurements, the coincidence between them is compelling. Where, for instance, Field, Lunardi and Settle found a measure of 42.55 cm, the computer established a dis-

178


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tance of 44.85 cm (Al to vii in Fig. 6). Where the measurements by hand established a distance of 19.05 cm, the computer determined the distance to be 19.49 cm (vii-vi in Fig. 6).38 Taken sin- gly, the distances calculated by the computer are interesting. Taken together as a related group of measurements (see additional measurements cited in note 38), they are strongly suggestive of the probability that Masaccio and Brunelleschi constructed the apparent diminution of the vault ribs using a relatively simple projection technique dependent on the surface grid and directly derived from the astrolabe.

The relationship between Masaccio's projection of the arched ribs of the Trinity and the astrolabe is clarified when the projection techniques found on the popular instrument are followed step by step. An illusionistic diagram of almucantars shows how the astronomer's system of celestial latitude planes (marked 100

- 800 in the upper section of Fig. 7) relates obliquely to the celestial equator when the observer is located at 360 north latitude. It further reveals how two separate sets of spatial coordinates (the first being the relationships among the Tropic of Capricorn, the Tropic of Cancer, and the celestial equator and the second being the relationship of the celestial latitudes and the celestial equator as determined from the position of the observer) are at once resolved in a geometrically controlled way on a single plane of representation (recorded in the lower section of Fig. 7).

To project the celestial latitudes onto the plane of the equator (the presumed plane of projection of the astrolabe), the instrument-maker had first to present the latitude circles in vertical section so that the observed horizon plane at 00 (the Horizon obliquus) is shown as a trace line and as a diameter bisecting the celestial plane [Fig. 8]. This planimetric configuration of the

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9) Second stage in the stereographic projection of almucantars. Originally drawn by B. Prestroud, Photographic Services, VPI & SU, Blacksburg, Virginia. Redrawn by Gorman.

celestial latitudes could well have been useful to Masaccio in placing the longitudinal ribs along the circumference of the Trin- ity arch at widths calculated by degrees.39

The planimetric section of the latitude system thus arrived at by astronomical instrument-makers was then rotated into its proper relationship with the plane of the celestial equator (Hori- zon rectus) when the latter was assumed to be at right angles to the plane of representation and thus was drawn as a trace line [Fig. 9].40 Lines extending from the projection point (commonly located at the south celestial pole) to the intersection (shown here as a trace line tangent to the north celestial pole) determined the diameters (and thus the centers and radii) of the planes of latitude ultimately projected onto the surface of the astrolabe [Fig. 10]. The projection process illustrated here shows that celestial latitudes were projected as a series of circles whose centers and radii had been transferred to the central vertical axis of the equatorial plane of the astrolabe. This process of transfer- ence would seem to have inspired Masaccio and Brunelleschi in their search for mathematical certainty.41

I have presented in separate steps what medieval treatises on the construction of the astrolabe commonly showed in a single drawing, where all the steps were integrated as in Masaccio's construction, with the central vertical axis representing simulta- neously the plane of projection perpendicular to the plane of representation and additionally a vertical division of the surface composition [Fig. 11]. Diagrams like that in Fig. 11 from a 14th- century copy of Messahalla's treatise on the astrolabe were a clear demonstration of the principles involved in developing information about curved spatial coordinates in a single drawing or representational field and could easily be used by painters and architects. This simultaneous presentation of projected planes rotated 900 with respect to the plane of representation was, however, pointedly rejected by Alberti, who drew the lateral section in a "separate space." By this separation Alberti freed the projection from the constraints of the circle and the square, something the astrolabist could not do and Masaccio was apparently unwilling to do.42

While the perspective projection of the Trinity "was easily constructed within the surface geometry," as Professor Polzer has maintained, that geometry-with its simple squares, semi- circles, and vertical axis; its centers and radii; its horizon line and siting point; its parallel and perpendicular relationships; and its trace lines and rotation of intersecting planes-derived from the live tradition of astronomical graphics, where the viewer and the viewed had long been subject to the "right reason" of geometrical analysis so admired by Alberti and where the integra- tion of the visual and the geometrical had been effected in a way directly useful to the early perspectivists and in keeping with their understanding of the highest and most inclusive truth.

180


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10) Third stage in the projection of almucantars showing the transfer of centers and radii and construction of latitude circles on the face of the astrolabe. Originally drawn by C. Joyce. Redrawn by Gorman.

In summary, with regard to the projection of the interior architectural space of the Trinity fresco, Masaccio sought instruc- tion from the many projection techniques enjoying particular popularity among late medieval astronomers because they recorded celestial phenomena with mathematical certainty. These refined and illusionistically suggestive constructions presented space according to a series of set coordinates specifically useful to the perspectivist and directly applicable to the Trinity fresco. In the astrolabe, planar coordinates were organized around a single point and, as in linear perspective, that point was associated with the act of seeing. In addition, it can be seen that many of the analytic conventions and construction techniques presented in medieval manuals on the astrolabe were critically important to the formulation of linear perspective. These include the definition of the projection surface as a plane, the distinction between the plane of projection and the plane of representation, the rotation of coordinate planes with respect to the plane of representation, and the systematic transfer of projected coordinates from one plane to another. Moreover, through a knowledge of astrolabic construction techniques, Masaccio and Brunelleschi could easily understand how the surface geometry of medieval paintings might be joined to a more strictly math-

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ematical definition of how points, lines, and planes function. This new articulation of the artist's basic formal means became explicit in Alberti's later treatise and is fundamental to the success of linear perspective.

All of us can be awed by the solemn humanity of Masaccio's images. Although few are likely to be stirred by the minute per- fections and careful consistencies of the perspective method he used to make this humanity possible, perhaps we can see how this early Renaissance artist applied to man and his earth-bound works the mathematical truths of a purposeful nature as he believed they existed and as they were presented to him by the ceaseless revolutions of the heavens.

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JANE ANDREWS AIKEN

This is a revised version of a paper presented at the annual meeting of the College Art Association, Boston, Mass., 1987. I wish to thank Lilian Armstrong, Peter Fergusson, Miranda Marvin, and Loren Par- tridge for their helpful suggestions. I also am immensely grateful for the guidance and support of James S. Ackerman to whom this essay is dedicated.

1 Aside from numerous incidental discussions in general texts, some studies devoted exclusively to the perspective projection of the Trinity fresco include: G. F Kern, "Das Dreifaltigkeitsfresco von S. Maria Novella, eine perspektivisch-architekturgeschichtliche Studie," Jahrbuch der K6niglich Preuzischen Kunstsammlungen XXXIV (1913), pp. 36-58; U. Schlegel, "Observations on Masaccio's Trinity in Santa Maria Novella," The Art Bulletin XLV (1963), pp. 19-33; J. Coolidge, "Further Observations on Masaccio's Trinity," The Art Bulletin XLVIII (1966), pp. 382-86; H. W. Janson, "Ground Plan and Elevation in Masaccio's Trinity Fresco" in Essays in the History of Art Presented to Rudolf Wittkower, London, 1967, pp. 83-88; J. Polzer, "The Anatomy of Masaccio's Holy Trinity," Jahrbuch der Berliner Museen XCIII (1971), pp. 18-59; J V. Field, R. Lunardi, and T. B. Settle, "The Perspective Scheme of Masaccio's Trinity Fresco, Nuncius IV.2 (1989), pp. 31-118. Among other excellent discussions of the Trinity are those of J. Mesnil, "Die Kunstlehre der FrOihrenaissance im Werk Masaccios, Vortrige der Bibliothek Warburg, 1925-26, pp. 122-146 and E. Hertlein, Masaccios Trinitit: Kunst, Geschichte und Politik der FrOhrenaissance in Florenz, Florence, 1979. Kim H. Veltman has written a forthcom- ing, impressively complete bibliography on perspective and its sources.

For Alberti's treatise On Painting of 1435/36, in which he for- mulates the rules and method of artist's perspective as deriving from the principles of Euclidean optics, see On Painting and On Sculpture: The Latin Text of De Pictura and De Statua, edited with translation, introduction, and notes by Cecil Grayson, London and New York, 1972. All subsequent references toDepictura andDe statua are to Grayson's edition.

2 On the assumption that the triptych from San Giovenale at Cascia di Reggello (dated 1422 by inscription) was executed by Masaccio, this work would be the earliest extant painting to display a "geometrically" correct perspective construction in that there is one centric point to which all orthogonals converge and a mathematically controlled diminution of the spaces between the horizontal lines (transversals) on the floor. Because of the high point of sight, however, the projected space of the painting is disturbingly acute, and the accu- racy of the placement of the transversals cannot be verified. The painting is perhaps best regarded as a partially successful perspective experiment deriving as much from Ambrogio Lorenzetti's achieve- ments of the late 1340's as from Brunelleschi's much discussed experiments involving a hand held panel and mirror. The latter would have been of little practical use to a painter working on either a five- foot long panel or a monumental fresco. If one accepts Decio Gioseffi's conclusion about the systematic regularity of the projection in the San Giovenale triptych, however, then one has to ask about the origin of that system? Is Masaccio already savvy about the most radical and theoretically complicated ideas concerning vision, space, and pictorial illusion investigated before 1422 by Brunelleschi and Donatello? See L. Berti, "Masaccio, 1422," Commentari 12 (1961), pp. 85-107 and D. Gioseffi, "Prospettiva,"Enciclopedia Universale dell' Arte XI, 1963, p. 141 and Tav. 120.

While Masaccio's Brancacci Chapel frescoes, especially the Tribute Money, show the artist to have been sensitive to many of the perceptual issues related to illusionistic fidelity, his abiding genius is

less consumed by the geometrical structure of a mathematically har- monious space as a reflection of divine order than with the awesome dignity of Christ and Peter. The recent cleaning of the chapel frescoes has shown that Masaccio used two different methods for projecting the building and steps on the right in the Tribute Money, thus creating a loosely constructed space in keeping with the narrative intent of the whole. The presence of two vanishing points gives clear evidence of Masaccio's willingness to experiment with projection techniques. See K. Christiansen, "Some Observations on the Brancacci Chapel Frescoes after their Cleaning," Burlington Magazine CXXXIII (1990), pp. 4-20.

Although the perspective construction of the Pisa polyptych of 1426 may have been as elaborately conceived as that of the Trinity fresco, the architectural setting of the throne with its predominant rectilinear form would not have presented the same problems for the artist as the fresco vault. J. Shearman, "Masaccio's Pisa Altarpiece: An Alternative Reconstruction," Burlington Magazine 108 (1966), pp. 449-55 and C. G. Von Teufel, "Masaccio and the Pisa Altarpiece: A New Approach," Jahrbuch der Berliner Museen XIX (1977), pp. 23-68.

3 The spatial peculiarities and inconsistencies of scale are sum- marized by Janson ("Ground Plan," passim.): the figures are not ren- dered di sotto in su; they are not in correct proportion with respect to their apparent position in the vaulted space; the first two rows of the vault coffers seem to be incorrectly foreshortened; and, most irksome to many (but not all) historians, God the Father cannot be properly located because His feet appear planted on a ledge near the rear of the vault, while His hands hold the cross, which is toward the front.

It is generally agreed that the centric point (also called a central vanishing point) is placed at the eye level of the average viewer, a level Janson ("Ground Plan," p. 85) determined to be approximately 153 cm from the church floor when the fresco was executed during the 15th century. Although other estimates of the height of the centric point range from 172 to 185 cm, an area about 2 cm across with its center at 172 cm has recently been established by the careful measurements of Field, Lunardi, and Settle ("Perspective Scheme," p. 37)

4 The fresco was covered by Giorgio Vasari's altar in the 16th century; moved to canvas in 1850; restored and relocated to its origi- nal site in 1950-54; and cleaned in 1969. As Polzer states ("Anatomy," p. 18): "closer inspection reveals many buckles and repairs, surely scars of the many moves and injustices the fresco has suffered."

5 Polzer, "Anatomy," p. 21 and passim. 6 Among the "diversity of means" Polzer identifies, excluding

the perspective of the vault, are the various construction techniques for foreshortening the dentil frieze, the capitals, the tondi, and the pseudo-meander. For a discussion of this unAlbertian mix of stereographic, orthographic, and planimetric projection techniques found in the fresco, see my Renaissance Perspective: Its Mathemati- cal Source and Sanction, unpublished doctoral dissertation, Harvard University, 1986 (University Microfilms International, Ann Arbor, 1986), particularly chapters IV and V. Other relevant discussions are those of B. B. Johannsen and M. Marcussen, '"A Critical Survey of the Theoretical and Practical Origins of Linear Perspective," Acta ad Archaeologium et Artium Historiam Pertinentia, Istitutem Romanum Norwegiae, Series Altera in 8, I (1981), pp. 191-229; M. Kemp, "Geo- metrical Perspective from Brunelleschi to Desaurges: A Pictorial Means or an Intellectual End?," Proceedings of the British Academy LXX (1984), pp. 89-132.

7 Bruneileschi's direct intervention in Masaccio's work provides a reasonable, though not historically verifiable, explanation for the

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technical sophistication and diversity of the space constructions not only in the Trinity, but also in the Pisa altarpiece. Few scholars doubt Brunelleschi's influence on Masaccio, and some posit his actual participation in executing the fresco.

Brunelleschi's assumed expertise is based on his reputation as the rediscoverer or discoverer or inventor of perspective (the particular epithet depending on whether one believes the Greeks pos- sessed a perspective projection or whether perspective is expressive of a psychophysical aspect of human perception). His competence supposedly issues from two demonstration panels that may antedate Masaccio's fresco by more than a decade. Many of the questions surrounding Brunelleschi's panels have been carefully considered by Kemp, "Science, Non-Science and Nonsense: The Interpretation of Brunelleschi's Perspective," Art History 1 (1978), pp. 131-61.

8 According to Polzer's measurements ("Anatomy," pp. 47-48), the width of the composition ranges from 210.5 to 211.8 cm. This measurement has been recently confirmed by Field, Lunardi, and Settle ("Perspective Scheme," p. 52) who determined the diameter of the arched molding, which can be taken as the diameter of the vault "as- suming that the picture plane coincides with the plane of the entrance arch," to be 211.55 cm or 3.625 Florentine braccia. However, Field, Lunardi, and Settle also found ("Perspective Scheme," p. 52, n. 39) that the distance between the outer edge of the columns was 205.1 cm or 3.514 braccia; this finding would seem to strengthen Janson's idea ("Ground Plan," pp. 83-84) that Masaccio had built up his composition in seven surface units of 1/2 braccio. The latter measurement also fits well with Polzer's contention that the sides of the top square of Masaccio's composition extend 207 and 206 cm on the left and the right respectively from the lower edge of the entablature to the Ionic capitals at the rear of the vault. Polzer argues that the middle square extends from just above the centric point and has a height of 207.3 cm. He believes the final square overlaps the middle one up to the level of the floor of the painted chapel sanctuary and is 204 cm high, the exact equivalent of 3 1/2 braccia.

That Masaccio would have scaled his composition in standard Florentine measurements is not only reasonable, it is also consistent with contemporary artistic practice as shown by D. F Zervas in her article "Ghiberti's St. Matthew Ensemble at Or San Michele: Sym- bolism in Proportion," The Art Bulletin 58 (1976), pp. 36-44. Although the Florentine artist's general interest in meticulous measurements and instruments of measure does not become explicit until Alberti's writings of the 1430s and 1440s, this interest is well established by the beginning of the century. Samuel Edgerton discusses this technical and conceptual revolution in "The Renaissance Artist as Quanti- fier," in The Perception of Pictures, ed. M. A. Hagen, New York and London, 1980, 2 vols., I: Alberti's Window: The Projective Model of Pictorial Information, pp. 179-211.

James Elkins has argued against Polzer's understanding of the surface geometry of the Trinity. See "The Case Against Surface Ge- ometry," Art History 14.2 (1991), pp. 161-65.

9 To the Gothic artist, the measuring and proportioning of a painting was a practical necessity for creating a level and square compositional field. See Cennino Cennini, The Craftsman's Handbook, trans. D. V Thompson, New York, 1954, chap. LXVII. In Chapter XXX, Cennini is particularly concerned with establishing a module that will tie together figure, building, and space.

Other uses of measure and geometry relating to structural sta- bility and symbolic efficacy are discussed by P Frankl, "The Secret of the Medieval Masons with an Explanation of Stornoloco's Formula by Erwin Panofsky," The Art Bulletin XXVII (1954), pp. 46-60; J. White,

"Measurement, Design and Carpentry in Duccio'sMaesth," Part I, The Art Bulletin LV (1973), pp. 356-58; D. F. Zervas, "The Trattato Dell'Abbaco and Andrea Pisano's Design for the Florentine Baptis- tery Door," Renaissance Quarterly XXVIII (1975), pp. 483-503.

10 Polzer, "Anatomy," pp. 47-48. 11 Alberti presents his perspective projection in De pictura,

1.19-20. 12 Rona Goffen, "Masaccio's Trinity and the Letter to the He-

brews," Memorie domenicane ns 11 (1980), pp. 498-99. Other scholars who have augmented our understanding of the secular, theological, and liturgical context of the Trinity include: Hertlein, Masaccios Trinitit, who gives the fullest account of the possible secular associations of this profoundly religious work, esp. pp. 177-84 and 193-95, 205-07; and 0. von Simson, "Uber die Bedeutung von Masaccios Trinitatsfresco in S. Maria Novella," Jahrbuch der Berliner Museen 6 (1966), pp. 119-59.

13 Vitruvius's formulaic image of the perfectly proportioned man contained in the circle and the square (De architectura, iii.l.3) remained popular during the entire 15th century. Although Leonardo da Vinci's drawing in the Venice Academy (R343) is perhaps the most famous rendering of this subject, a lesser known sketch by the Sienese engi- neer Mariano Taccola from the early 1430s is an equally direct expression of how perfect geometrical forms generate the proportions of "Man as Microcosm." See F D. Prager and G. Scaglia, Mariano Taccola and his Book De Ingeneis, Cambridge, MA, 1972, pp. 42, fig. 7 and 167-69.

Unlike Vitruvius's rationally antiseptic formulation, however, numerous medieval images link the cosmic man quite literally to both Christ and to the Christian Cosmos. This connection was a subject of much interest to medieval mystics like Hildegard of Bingen, as H. Torp has shown in "The Integrating System of Proportion in Byzantine Art, An Essay on the Method of the Painters of Holy Images," Acta ad Archaeologiam etArtium Historiam Pertinentia, Series in Altera 8, IV (1984), pp. 134-47, Figs. 39-43. Torp (p. 140) notes that "the height of [Hildegard's] microcosmic man corresponds to three times the diameter of the earth." This, according to Hildegard, is "the form in which Christ will reveal himself on the Day of Judgment (Torp, p. 146)." See below, for the possible significance of such cosmic triades to Masaccio.

Other discussions of the common associations existing among art, mathematics, and religion are found in R. Wittkower, Architec- tural Principles in the Age of Humanism, London, 1952, particularly pp. 24-27 and S. Y. Edgerton, Jr., The Renaissance Rediscovery of Linear Perspective, New York, 1975, pp. 16-21.

14 Polzer ("Anatomy," p. 45) gives measurements for two indented centric points located in a restored area in the center of the fresco just below the step on which the donors kneel. According to his conclusions, the centric point is placed between 174 cm and 177.5 cm from the floor. According to the more recent measurements of Field, Lunardi, and Settle ("Perspective Scheme," p. 37), the centric point is in the center of a circle 2 cm in diameter at 172 cm above the floor; this latter measure is less than a 2% skew from being precisely three standard Florentine braccia from the floor (175.09 cm).

There are at least four reasons why all measurements of the fresco might not coincide precisely with braccia measures: the fresco was moved, the floor of the church was repaved in the 19th century, the areas measured have been restored, and Masaccio made numerous adjustments to the measurements to achieve a visual and aesthetic resolution that continues to awe us. Nonetheless, Masaccio's probable use of local standard measurements in proportioning his surface grid and in determining the position of the centric point is

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consistent with Florentine workshop practice of the 1420s. One may assume with some confidence that Alberti's recommendation of 1435/ 36 about paintings being scaled tobraccia units was prompted by his artist friends (De pictura 1.19). The height Alberti recommends for his "well-proportioned man" in both De pictura and De statua to express the "perfect beauty distributed by Nature" is three braccia.

The "average" height Alberti speaks of and Masaccio apparently uses in the perspective scheme of the Trinity should be regarded as denoting not only the typical in the quantitative sense but also, and more emphatically, the ideal and essentially true in the qualita- tive sense. See R. Wittkower, "The Changing Concept of Proportions," Daedalus 89 (1960), pp. 199-215; P H. Michele, "L'esth6tique arithm6- tique du quattrocento: une application des m6di6t6s Pythagoriciennes %

I'esth6tique architecturale," M6langes offerts a Henri Houvette, Paris, 1934, pp. 181-89.

That Masaccio distinguished between the viewer standing in front of the fresco and the truly average man is suggested by the height of the recumbent skeleton beneath the vaulted space. Janson ("Ground Plan," p. 85) determined the height of this man to be 160 cm. Redemption is potentially his; however, he cannot stand for the perfect man, and his height is one indication of that lesser status. See C. Dempsey, "Masaccio's 'Trinity': Altarpiece or Tomb?," The Art Bulletin 54 (1972), pp. 279-81.

Field, Lunardi, and Settle ("Perspective Scheme," pp. 57-58) found the painted height of God Father to be 155 cm and, because of his distance from the picture plane, assumed his "true" height to be 3 braccia. This consonance of measurements would seem to be simply another aspect of the Trinity which, as Rona Goffen has asserted ("Masaccio's Trinity," pp. 497-98), shows Masaccio equating the principal viewer of the fresco, with the priest who celebrates the Eucha- rist, this priest being God Father as the Person of the Trinity who presents the Son as Eucharist. Martin Kemp would seem to agree with the possibility of a consonance existing between the three braccia measure and the figure of Christ as he inserts a much reduced photo- graph of Masaccio's Christ into the scaled units of Alberti's perspective projection (The Science of Art, New Haven and London, 1990, plate 24).

15 Zervas ("Trattato," n. 24) speaks of the religious significance of the 3braccio measure as follows: "According to G. Uzielli, Le misure lineari mediaeval e I'effigie di Cristo, Florence, 1899, p. 13, the Florentine braccio was derived from the legal Palestinian braccio, which was introduced to the West during the Crusades of the twelfth and thirteenth centuries. The Palestinian braccio was adopted as a unit of measure in Pisa in the twelfth century because of its special religious significance; It was used to calculate the length of the body of Christ (3 Palestinian braccia). This is verified by a number of Trecento and Quattrocento guidebooks to the Holy Land, which contained a linear measure identified as 1/6 the length of Christ's body: the measure is usually a subdivision of the Palestinian braccio (55.48 cm)." Certainly Alberti was aware of this tradition since, in De statua, he divides the perfect man into six units called exempeda.

One need not rely, however, on guidebooks to Jerusalem to ascertain the mensura Christi; four columns formerly in the aula del Concilio, the Lateran, Rome, were believed in Masaccio's time to es- tablish the height of Christ at approximately 1.78 cm. This measurement, according to C. Ginzburg (The Enigma of Piero, trans. M. Ryle and K. Soper, London, 1985, pp. 129-30), was familiar to Piero della Francesca when he calculated the height of Christ in his Flagellation. Certainly the Dominican monks of Masaccio's time would have been aware of this traditional measurement. This tradition adds, what may

be an unwanted flexibility to the situation, since then the 172 cm height of Masaccio's centric point would represent the eye level of Christ, rather than the height. In any event, the centric point of the Trinity is much too high to represent the height of the average Florentine. That "Christ is both sacrifice and priest," as Goffen has said ("Masaccio's Trinity," p. 504), is confirmed here in the typically Renaissance dependence on measure and quantity.

16 For the special Dominican concern for the cult of Corpus Domini, see Eve Borsook, The Mural Painters of Tuscany From Cimabue to Andrea del Sarto, 2nd revised ed., Oxford, 1980, pp. 49 and 58; R. Trexler, "Florentine Religious Experience: The Sacred Im- age," Studies in the Renaissance XIX (1972), pp. 9-11.

17 On the assumption that either Masaccio or Brunelleschi might have sought an acceptable explanation for how geometry met the demands of both theology and nature, he would have found it in the prologue to one of the most popular astronomical texts of the time, Campanus of Novara's Theorica planetarum: "The foremost master of philosophy divides the provinces of that [subject] into three primary genera; the first of these he names theological, the second mathematical, and the third natural. The middle term becomes in a way a partaker in the nature of the two extremes, because mathematical principles are found in the realm of nature and theology alike." F S. Benjamin and G. J. Toomer, Campanus of Novara and Medieval Plan- etary Theory, Theorica planetarum, Madison-Milwaukee-London, 1971, pp. 136-39.

18 The continued influence of Euclid and the Euclidean tradition hardly requires comment. The many projection techniques of Ptolemy's treatises (Annelemmate, Geographia, and Planisphaerium) were fundamental to the graphic tradition of medieval astronomers, map-makers, and instrument makers. For the possible influence of Ptolemy's Geographia on early perspective experiments see, Edgerton, Renaissance Rediscovery, pp. 91-123. R. B. Thomson (Jordanus de Nemore and the Mathematics of Astrolabes: De Plana Spera, edited with an introduction, translation and commentary by R. B. Thomson, Toronto, 1978, pp. 53-54) notes that by the 13th century, Messahalla's treatise on constructing the astronomical siting device known as an astrolabe had been expanded "into a sort of corpus" of construction and projection methods particularly appropriate for in- struction.

For an evaluation of ancient mathematical diagramming conventions, see Aiken, "Perspective," chapters I and II. The historians of mathematics most strongly engaged in this subject include O. Neugebauer, A History of Ancient Mathematical Astronomy, 3 vols. Berlin-Heidelberg-New York, 1975; W. Knorr, The Evolution of Euclid- ean Elements: A Study of the Theory of lncommensurate Magnitudes and Its Significance for Early Greek Geometry, Boston and Dordrecht, 1975; J. E. Murdoch, Album of Science: Antiquity and the Middle Ages, New York, 1984; and B. S. Eastwood, Astronomy and Optics from Pliny to Descartes: Texts, Diagrams and Conceptual Structures, London, 1989.

For the most part, only in the past twenty years have historians of mathematics and art begun to lay the groundwork for a serious investigation of the graphic conventions employed by various technical and scientific disciplines during the medieval period which may have been instructive sources to the early perspectivists. In addition to Edgerton's first decisive steps in 1975, a review of some earlier studies is found in Johannsen and Marcussen, "Critical Survey." Also, Kemp ("Science") has considered the probable influence of medieval surveying techniques on Brunelleschi's perspective experiments. Jehane R. Kuhn, "Measured Appearances: Documentation and De-

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sign in Early Perspective Drawing," Journal of the Warburg and Courtauld Institutes 53 (1990), pp. 114-32, has thoughtfully evaluated the practical, theoretical, and conceptual environment within which Brunelleschi's perspective schema arose and their dependence on the technical culture of early fifteenth-century Florence. In addition to Zervas's comments on an artist's education ("Trattato Dell'Abbaco"), see Gino Arrighi's articles on the artisan's education in the exact sciences; these include "Le scienze esatte al tempo del Brunelleschi," in Filippo Brunelleschi: La sua opera e il suo tempo, pp. 93-103, ed. G. Spadolini, Florence, 1980 and "La matematica del Rinascimento in Firenze. Leredita di Leonardo Pisano e le 'botteghe d'abbaco'," Cultura e Sculoa 18 (1966), pp. 287-94.

Most pertinently, in my view, K. H. Veltman has lectured on the points of correspondence existing between Alberti's perspective construction and Ptolemy's method of projection ("Ptolemy and the Origins of Linear Perspective," in La prospettiva rinascimentale: Codificazione e trasgressioni, I, ed. M. D. Emiliani, Milan, 1980, pp. 403-07.) See also Veltman's review of Edgerton's Renaissance Redis- covery in the The Art Bulletin LIX (1977), pp. 28-82.

19 Although there is a great deal of speculation about who might have instructed Brunelleschi in mathematics, as a trained goldsmith and someone interested in clock making, Brunelleschi would have known how to execute all the basic projections found on the astrolabe (See F D. Prager, " Brunelleschi's Clock?," Physis 10 (1968), pp. 201-16). Brunelleschi's interest in surveying also would have taught him some of the down-to-earth uses of the astrolabe. Finally, the creative architect would have known the architectural rendering practices of his day, including such intricacies as constructing a 'quinto acuto' arch with "radii equal to four-fifths of the base diameter of the arch with respective centers at the four-fifths points on either side of the base diameter." The reason why I quote H. Saalman's foregoing description of a construction technique known well to Brunellleschi is that, while the construction may seem arcane to the uninitiated, it was understood by Brunelleschi and his contemporaries. I submit that same is true of astrolabic projections. See Saalman, " Giovanni di Gherardo da Prato's Designs concerning the Cupola of Santa Maria del Fiore in Florence," Journal of the Society of Architectural Histori- ans 18 (1959), p. 16. Of interest also with respect to Brunelleschi's understanding of technically advanced diagramming protocols and his education are: G. Arrighi, II codice L.VI.21 della Biblioteca degli Intronati di Siena e la 'Bottega dell' Abbaco' a Santa Trinith in Firenze," Physis VII (1965), pp. 369-400; R Sanpaolesi, "Le conoscenze technica del Brunelleschi," inFilippo Brunelleschi: La sua opera e i/suo tempo, Florence, pp. 150-57.

20 In the projection techniques of astronomers from before Ptolemy and on into the 15th and 16th centuries, points were assumed to have the capacity to mark the position of lines or rays extending perpendicularly to the plane of projection (Thomson, Jordanus de Nemore, pp. 28-29). Any person who picked up an astrolabe would understand that a point signified a line of sight. Alberti, for instance, is simply stating a generally accepted practice of medieval astronomers and their instrument makers in De pictura 1.19 when he says of the centric point: "...it occupies the place where the centric ray strikes."

21 For the function of trace line projection in the ancient diagrams of sterea (three dimensional bodies), see Neugebauer, Ancient Math- ematical Astronomy, pp. 860-61. In the diagram of the cosmos, the primary reason for projecting planes as trace lines was to document the exact juncture of plane to plane and of plane to axis in the revolv- ing spheres of the universe. All medieval diagrams of celestial phenomena use trace lines in the same way as their ancient forebears.

22 The principle that the arc of a curved plane (the plane of the observed horizon) appears as a straight line when in the same plane as the eye (and thus perpendicular to the plane of representation according to the specific conditions of artist's perspective) was established by Proposition 22 of Euclid's Optics. Since neither Brunelleschi nor Masaccio had read Euclid, this principle must have been a com- monplace of the day.

23 Although the rotation of planes in Masaccio's fresco is not unlike Brunelleschi's method of projection in his experimental panels (perhaps dated to as early as 1413) and as characterized by R. Krautheimer and T. Krautheimer-Hess (Lorenzo Ghiberti, 2 vols., Princeton, 1970, pp. 237-40), the principle and method of rotating the plane of projection 900 with respect to the plane of representation was so firmly established in contemporary astronomical graphics that they were not even discussed by late medieval astronomers (Thomson, Jordanus de Nemore, pp. 58-60).

That Masaccio found it necessary to scratch the vertical axis of his composition into the plaster for technical reasons unrelated to his perspective construction is not questioned here. Aside from being an integral part of the perspective construction, the axis line was simply a means for keeping the composition level and square and for aiding in the transfer of drawn models to the wall surface.

24 While artists would have been schooled in the understanding of the function of the plane of projection through less elevated means, an unequivocal presentation of these essential assumptions of ancient diagrammatic techniques is found in the writings of Archimedes, particularly The Sand Reckoner (The Works of Archimedes with the Method ofArchimedes, ed. T. L. Heath, Cambridge, pp. 222-26). Other relevant comments on the nature of mathematical diagrams by Archimedes appear in On Conics and Spheroids, Propositions 23 and 27-30. Archimedes's writings on conics were especially important to the 13th- and 14th-century scholars who studied optics, the science that seems to have been centrally important for energizing the dy- namic intellectual and cultural environment out of which artist's perspective developed.

25 K. Veltman has previously concluded ("Ptolemy," p. 407) that "the context of perspective is to be found amongst the many off- spring of Mother geometry and in particular her daughter, astronomy." Johannsen and Marcussen ("Critical Survey" pp. 217-23) have discussed some of the ways astronomy coincides with the most noteworthy characteristics of perspective projection. G. ten Doesschate (Perspective Fundamentals, Controversials, History, Nieuwkoop, 1964, pp. 143-54 and n. 17) suggested some connec- tions between art and astronomy, and Field, Lunardi, and Settle (Per- spective Scheme," p. 115) have concluded that "the form of projection used in almost all European astrolabes in Brunelleschi's day is mathematically equivalent to the projection used in perspective construction."

Despite the conclusions of a few, art historians have overlooked or resisted the possibility that the early students of perspective could have understood the intricacies of astrolabic projection without the "active participation of experts as interpreters" (Johannsen and Marcussen, pp. 226-27).

In a culture where one told time by the stars and sailed by them, practiced medicine in accord with them and set major religious feasts in consonance with them, planted crops and prognosticated the future according to their configuration, it is beyond the realm of possibility that the same man who designed the great dome of the cathedral of Florence did not understand the astrolabe and its many layered projections.

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Historians of science regard knowledge of astrolabic projection as eminently available. H.L.L.Busard ("The Practica Geometriae of Domenicus of Clavasio," Archive for the History of Exact Science II (1965), p. 522) notes that Domenicus of Clavasio omits a description of the quadrant and the astrolabe in this most basic of textbooks because these would be "sufficiently known." W. Hartner ("The Prin- ciple and Use of the Astrolabe," in A Survey of Persian Art from Pre- historic Times to the Present, VI, pp. 3533-3534, A. U. Pope and

P. Ackerman, eds., Oxford, 1938-39, reissued 1967) comments on the widespread use of the astrolabe. J. D. North ("The Astrolabe," Scien- tific American CCXXX (1974), p. 96) categorically states: "The astrolabe was the most widely used astronomical instrument of the Middle Ages." Even Chaucer signals the common enough possession of the astrolabe, for his "poure scoler" from the "Miller's Tale" possesses "His Almageste, and bookes grete and smale His astrelabie..."

Chaucer's little manual on the astrolabe, Bread and Milk for Children, again alerts us to the fact that what we have come to imag- ine as difficult to comprehend was considered in the late 14th century a simple matter appropriate for the education of the young. Any ordinary academic walking the streets of Florence would have been up to the task of explaining some unresolved complexity of astrolabic projection for an artist. Further, as L. White, Jr. has pointed out ("Medi- cal Astrology and Late Medieval Technology," in Medieval Religion and Technology, Collected Essays, Berkeley, Los Angles, and London, 1978, pp. 301-11) many late medieval academics tinkered with astronomical instruments, suggesting that applied science and astronomical graphics were equally vital topics in some minds. For a more complete discussion of the relationship existing among late medieval astronomical instruments, applied science, and artist's perspective see my "Truth in Images: From the Technical Drawings of Ibn al- Razzaz al-Jazari, Campanus of Novara, and Giovanni de'Dondi to the Perspective Projection of Leon Battista Alberti," Viator 25 (1994), pp. 325-359.

26 For a discussion of Alberti's perspective construction and how it depends on, differs from, and refines that of the Trinity fresco, see Aiken, "Perspective," chapter IIl.

27 For the highly suggestive language of medieval astronomical texts, see Doesschate (Fundamentals, pp. 143-54) and Thomson (Jordanus de Nemore, pp. 146-50).

28 North ("Astrolabe," p. 100) describes the principal plate of the astrolabe and its projections as follows: "It is graduated with a series of circles representing the coordinate lines that are fixed with respect to a given observer. The center of the astrolabe, around which the rete [a superimposed, fretted plate] turns, represents the north celestial pole, around which the stars appear to turn. Concentric with it are the Tropic of Cancer, the celestial equator and the Tropic of Capri- corn... On the plate there is a line representing the observer's horizon and the point of his zenith. There is a set of almucantars, or circles of constant altitude, above the horizon and encircling the zenith. There are also lines of constant azimuth, which appear as arcs of circles radiating from the zenith and running down from the horizon."

K. Veltman ("Ptolemy," p. 406) correctly observed that the azimuth and almucantar systems were projected "relative to the viewer's position." He goes on to say that they thus represent an "apparent point of view" (which is correct only in so far as the position of the "viewer" is calculated by determining his distance from the celestial equator). Despite the illusionistic appeal of the azimuth and almucantar systems [See Figs. 4 and 6], however, the "apparent view" of the astrolabe does not represent the perceived foreshortenings experienced by an astronomer gazing up at the heavens. It is, nonetheless, visu-

ally seductive and may have caught Brunelleschi's attention for that reason.

Another example of Brunelleschi's connection to astronomical images (though it is an ex post facto example) is the painted zo- diac in the dome over the altar of the architect's Old Sacristy, San Lorenzo. See Murdoch, Album, p. 249 and J. Beck, "Leon Battista Alberti and the 'Night Sky' at San Lorenzo," Artibus et historiae 19 (1989), pp. 9-34.

29 L. D. Ettlinger, "Pollaiuolo's Tomb of Sixtus IV," Journal of the Warburg and Courtauld Institutes XVI (1953), pp. 258-61.

30 E. Male, Chartres, New York, 1983, Figs. 27-28. where four, half-length figures holding astrolabes are clearly divine intelligences, not astronomers. L. White ("Temperantia and the Virtuousness of Technology," in Medieval Religion and Technology, Collected Essays, Berkeley-Los Angeles-London, pp. 181-204) was particularly concerned with the astronomical clock, which usually included an astrolabe among its dials and became a prime attribute of Temperance during the 14th and 15th centuries. At this time she was newly associated with Sapientia and the Logos. White's point is that instruments of measure were perceived as operating at a high moral level. Masaccio and Brunelleschi were no doubt aware of this common enough association between ingenious measuring devices and divine order.

31 Quoted from F Winwar's translation of Boccaccio'sDecameron VI.v by B. Cole, Giotto and Florentine Painting 1280-1375, New York, 1976, p. 162.

32 According to O. Neugebauer (The Exact Sciences in Antiq- uity, 2nd edition, New York, 1969, p. 219), the very problem to be solved in stereographic projection "consists in the determination of the centers and radii of the circles which are images of the circles on the celestial sphere when projected from the south pole onto the equator."

33 Polzer, '"Anatomy," pp. 54-55. 34 By making the central vertical axis of the composition signify

the intersecting element of the projection, Masaccio devises a technique differing significantly from that described by Alberti inDe pictura 1.19. The humanist determined the position of transverse elements from a second drawing called anareola. However, Masaccio's method of relying on the central axis of the compositional field itself, and thus having the axis function as a profile of the plane of projection, not only provided another direct parallel with astrolabic projection techniques, but also makes Giorgio Vasari's characterization of linear perspective "con pianta e profilo e per via della intesegrazione" seem particularly condign. See Le vite de'piu eccellenti pittori, scultori ed architettori II, Florence, 1879, p. 332.

35 Most discussions of the perspective projection of the Trinity, focus on the exact dimensions of the vaulted sepulchral space. If in figuring out how to project a convincing illusion of space for the special viewing conditions provided by the Trinity Brunelleschi and Masaccio determined the critical importance of the eye point (as Edgerton, Renaissance Rediscovery, as well as Field, Lunardi, and Settle, "Perspective Scheme" have argued), they may have left to Alberti the task of devising a mathematically consistent method for projecting the ground plane.

36 The measurements of the total height (A-E) and width (B-B2), and distances Al-B1, B1-C1, C1-D1, and D1-E1 in Fig. 6 are found scattered throughout Field, Lunardi, and Settle's essay on the perspective scheme of Masaccio's Trinity.

37 The most controversial distance generated by this astrolabic method of projection is the putative distance of the viewer (D1 to D3

186


or 218.9 cm). If Masaccio had not extensively adjusted some of the angles of the receding planes, as he did, and if he had not blocked one's view of the more dramatically foreshortened areas of the vault by placing the imposing sacred figures directly before one's eyes, then this short viewing distance would have resulted in a nearly im- possible viewing situation.

Field, Lunardi, and Settle have argued strenuously for a minimum viewing distance equal to the width of the side aisle, 686.25 cm, because "it looks right." While they begin with the assumption that this minimum distance is the proper one and then attempt to elicit that figure from the measurements taken from the fresco, their assumption cannot be proven. In fact, Field, Lunardi, and Settle, dem- onstrate beyond a shadow of a doubt that even a minor adjustment to the width of an abacus, for instance, will result in widely divergent viewing distances. The possible viewing distances admitted to by Field, Lunardi, Settle range from 345.66 cm to 721.3 cm.

38 Field, Lunardi, and Settle display their measurements in Table 2 of their essay ("Perspective Scheme," p. 96). The coincidence between their measurements and those in Fig. 6 are as follows:

F/L/S Table 2 Computer Diagram [Fig. 6]

A-C 42.55 cm Al-vii 44.85 cm C-E 19.05 cm vii-vi 19.49 cm E-I 16.45 cm vi-v 16.79 cm I-R 15.30 cm v-iv 16.04 cm

It should also be of interest to art historians that the distance Al-vii, which because of its length has usually been regarded as indicating a construction error on Masaccio's part in foreshortening the coffers, now would seem to be the result of the projection technique he employed. Also, if the measurements are convincing, they would put to rest another question about the structure of the vault, here shown to have nine curved ribs.

39 Field, Lunardi, and Settle ("Perspective Scheme," pp. 49-54 and 98-99) discuss some alternative methods Masaccio might have employed in determining the arcal distances of the longitudinal ribs. Compare, however, the suggestive consonance existing between Fig. 8 and the semi-circular area between i and C1 in Fig. 6 diagramming the rear wall of the vault.

40 The distinction made by astronomers between the "observed horizon" and the "celestial horizon" is an important one for painters. For an additional lesson that painters might have learned about the integral configuration of eye point, site line, and horizon line by study- ing the shadow squares inscribed on the back of most astrolabes and used by surveyors to measure the height of buildings, see Aiken, "Per- spective," pp. 223-29.

41 In effect, the transfer of centers in both astrolabe and fresco amounts to a rotation of the plane of projection-first drawn as a trace line and then rotated until parallel to the plane of representation. Again, it was Alberti who discussed both the fact and the theoretical implica- tion of this rotation.

42 Alberti, Depictura 1.20.

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