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Fig. 1. Facade of Milan Cathedral
Fig. 1. Facade of Milan Cathedral
Fig. 2. Survey plan of Milan Cathedral (after Ferrari da Passano, ‘Storia della veneranda fabbrica’, 1973)
Fig. 2. Survey plan of Milan Cathedral (after Ferrari da Passano, ‘Storia della veneranda fabbrica’, 1973)
was his intervention necessary? Frankl, like Camillo Boito and Luca Beltrami before him, insisted on what had been known since the ancient world: that the height of an equilat- eral triangle is incommensurable with respect to its base or side, but equals the base multiplied by \/3/2, to use modern mathematical notation. Frankl therefore asked how the master-masons could have used a system of equilateral triangles if they were only able to draw them geometrically but were not capable of calculating their heights. Having posed the problem in this form, he then proposed that Stornaloco had been summoned to Milan to calculate these dimensions.°® a
was his intervention necessary? Frankl, like Camillo Boito and Luca Beltrami before him, insisted on what had been known since the ancient world: that the height of an equilat- eral triangle is incommensurable with respect to its base or side, but equals the base multiplied by \/3/2, to use modern mathematical notation. Frankl therefore asked how the master-masons could have used a system of equilateral triangles if they were only able to draw them geometrically but were not capable of calculating their heights. Having posed the problem in this form, he then proposed that Stornaloco had been summoned to Milan to calculate these dimensions.°® a
Fig. 4. Mathes Roriczer, illustrations showing how to design a pinnacle by ‘constructive geometry’, starting from a square (from Biichlein von der Fialen Gerechtigkeit, 1486)
Fig. 4. Mathes Roriczer, illustrations showing how to design a pinnacle by ‘constructive geometry’, starting from a square (from Biichlein von der Fialen Gerechtigkeit, 1486)
was the role of practical geometry. Its importance is demonstrated by the fact that amongst the first books to be printed were the late fifteenth- and early sixteenth- century design handbooks by Mathes Roriczer, Hans Shmuttermayer and Lorenz Lechler, which provide practical instructions, accompanied by drawings, illustrating some of the steps needed for a correct design of some architectural elements, steps which at first sight might seem to require mathematical calculation (Fig. 4).°° In fact, however, their instructions come in the form of easy practical rules, based on simple geo- metric figures, which can be employed following a prefixed set of steps, without any need for analytical reasoning or formulae.*!
was the role of practical geometry. Its importance is demonstrated by the fact that amongst the first books to be printed were the late fifteenth- and early sixteenth- century design handbooks by Mathes Roriczer, Hans Shmuttermayer and Lorenz Lechler, which provide practical instructions, accompanied by drawings, illustrating some of the steps needed for a correct design of some architectural elements, steps which at first sight might seem to require mathematical calculation (Fig. 4).°° In fact, however, their instructions come in the form of easy practical rules, based on simple geo- metric figures, which can be employed following a prefixed set of steps, without any need for analytical reasoning or formulae.*!
Whilst there were discussions about various elevations ad triangulum or ad quadratum (i.e. schemes based on equilateral triangles or squares) there is no record of the existence of a model embodying any kind of elevational system before Stornaloco’s contribution of 1391. The construction of a model always demonstrated the seriousness of a project given the time and money involved, and the likelihood is that, as we shall now see, the first wooden model of the cathedral was made so as to incorporate Stornaloco’s scheme, thus indicating its absolute importance for the authorities. | if |, Se |) i : eS a ) , a ) a a a a a |
Whilst there were discussions about various elevations ad triangulum or ad quadratum (i.e. schemes based on equilateral triangles or squares) there is no record of the existence of a model embodying any kind of elevational system before Stornaloco’s contribution of 1391. The construction of a model always demonstrated the seriousness of a project given the time and money involved, and the likelihood is that, as we shall now see, the first wooden model of the cathedral was made so as to incorporate Stornaloco’s scheme, thus indicating its absolute importance for the authorities. | if |, Se |) i : eS a ) , a ) a a a a a |
symptoms of the incompetence of Milanese masters in comparison with Northern archi- tects (see Section 5). But the solution is far from convincing because it does not satisfy a fundamental requirement of the elevation of the project: the generation of the key points for constructing the ‘pointed-third’ vaults universally employed in Milan Cathedral. By contrast, Stornaloco’s system based on equilateral triangles (and rectangles) allowed the architects to generate without difficulty the curvature employed for all the vaults in the cathedral, specifically described by Cesariano as being of ‘pointed third’ (terzo acuto) shape.°” In fact, once the distance between the springing-points of the vaults of the nave and aisles was established, all the master-masons had to do in order to generate the required profiles for the vaults was to draw segments of two equal circles, each centred at the ends of the springing line of the vault and, using a radius equal to the span of the vault, make the segments intersect at the apex of the equilateral triangle. By contrast, Beltrami’s Pythagorean triangles 12 braccia high do not permit the construc- tion of pointed-third vaults, and, in fact, his hypothesis appears to rest on mistaken premises. The widths of the triangles that Beltrami proposed were generated by the interaxial distances between the piers (16 and 32 braccia), but their heights — to judge from his celebrated drawing (Fig. 7) — evidently refer to the intradoses of the vaults rather than their extradoses; and this is an error of logic since triangles based on interaxial distances should have heights corresponding to their extradoses and not their intradoses.°® a *@ Se + tt @ . owe
symptoms of the incompetence of Milanese masters in comparison with Northern archi- tects (see Section 5). But the solution is far from convincing because it does not satisfy a fundamental requirement of the elevation of the project: the generation of the key points for constructing the ‘pointed-third’ vaults universally employed in Milan Cathedral. By contrast, Stornaloco’s system based on equilateral triangles (and rectangles) allowed the architects to generate without difficulty the curvature employed for all the vaults in the cathedral, specifically described by Cesariano as being of ‘pointed third’ (terzo acuto) shape.°” In fact, once the distance between the springing-points of the vaults of the nave and aisles was established, all the master-masons had to do in order to generate the required profiles for the vaults was to draw segments of two equal circles, each centred at the ends of the springing line of the vault and, using a radius equal to the span of the vault, make the segments intersect at the apex of the equilateral triangle. By contrast, Beltrami’s Pythagorean triangles 12 braccia high do not permit the construc- tion of pointed-third vaults, and, in fact, his hypothesis appears to rest on mistaken premises. The widths of the triangles that Beltrami proposed were generated by the interaxial distances between the piers (16 and 32 braccia), but their heights — to judge from his celebrated drawing (Fig. 7) — evidently refer to the intradoses of the vaults rather than their extradoses; and this is an error of logic since triangles based on interaxial distances should have heights corresponding to their extradoses and not their intradoses.°® a *@ Se + tt @ . owe
vaults, where they simply placed their formwork at the pre-established height of 28 braccia. But the question then emerged when they had to start building the inner-aisle vaults and this was the subject of the discussions of 1392, when they finally ratified the changes of dimensions incorporated in the second scheme, Stornaloco II. However. when we examine modern survevs of the cathedral more clocelv (Tables 7 raccia. But the question then emerged when they had to start building the inner-aisle vaults and this was the subject of the discussions of 1392, when they finally ratified he changes of dimensions incorporated in the second scheme, Stornaloco II. However, when we examine modern surveys of the cathedral more closely (Tables 1 ind 2), it appears that Valentini’s limpid explanation, which in principle remains convin- ing, still requires some amendment. First, Valentini assumed that the clear-distance etween the piers was equal to the clear distance between the springing-points of the vaults, but this is not the case in practice. The clear distance between the nave-piers is ibout 28 braccia generating a height for the vault ad triangulum of 24 braccia, but the lear distance between the aisle-piers is about 12 braccia,°* and not the 14 braccia indi- ‘ated by Valentini, since the piers are all slightly broader than 4 braccia.°> If we were o follow Valentini’s hypothesis using the distances between the piers, the heights of he aisle-vaults generated by equilateral triangles would then be about 10 braccia, creat- ng a sequence of dimensions of 28 braccia for the imposts of the outer aisle vaults, 38 raccia for the imposts of the inner aisle vaults, 48 braccia for the imposts of the nave- vaults, and 72 braccia for the nave vault — dimensions that have nothing to do with he cathedral as built. However, Valentini’s explanation can be made workable if nstead one considers the clear-space between the springing-points of the vaults and 1ot the clear-space between the piers, because the springing-points are set back with espect to the vertical faces of the piers. Thus, the width between the springing-points yf the vaults of the outer aisle is in fact about 14 braccia and the height is about 12
vaults, where they simply placed their formwork at the pre-established height of 28 braccia. But the question then emerged when they had to start building the inner-aisle vaults and this was the subject of the discussions of 1392, when they finally ratified the changes of dimensions incorporated in the second scheme, Stornaloco II. However. when we examine modern survevs of the cathedral more clocelv (Tables 7 raccia. But the question then emerged when they had to start building the inner-aisle vaults and this was the subject of the discussions of 1392, when they finally ratified he changes of dimensions incorporated in the second scheme, Stornaloco II. However, when we examine modern surveys of the cathedral more closely (Tables 1 ind 2), it appears that Valentini’s limpid explanation, which in principle remains convin- ing, still requires some amendment. First, Valentini assumed that the clear-distance etween the piers was equal to the clear distance between the springing-points of the vaults, but this is not the case in practice. The clear distance between the nave-piers is ibout 28 braccia generating a height for the vault ad triangulum of 24 braccia, but the lear distance between the aisle-piers is about 12 braccia,°* and not the 14 braccia indi- ‘ated by Valentini, since the piers are all slightly broader than 4 braccia.°> If we were o follow Valentini’s hypothesis using the distances between the piers, the heights of he aisle-vaults generated by equilateral triangles would then be about 10 braccia, creat- ng a sequence of dimensions of 28 braccia for the imposts of the outer aisle vaults, 38 raccia for the imposts of the inner aisle vaults, 48 braccia for the imposts of the nave- vaults, and 72 braccia for the nave vault — dimensions that have nothing to do with he cathedral as built. However, Valentini’s explanation can be made workable if nstead one considers the clear-space between the springing-points of the vaults and 1ot the clear-space between the piers, because the springing-points are set back with espect to the vertical faces of the piers. Thus, the width between the springing-points yf the vaults of the outer aisle is in fact about 14 braccia and the height is about 12
Table 2. Principal heights of Milan Cathedral according to the Stornaloco I and Stornaloco II schemes compared with those given in modern surveys of the building. Dimensions in braccia in italics are converted from metres, but using decimals for convenience; dimensions in metres in italics are calculated from unannotated drawings. a passage of a well-known document recording the general meeting of 1 May 1392 (Appendix II, doc. I). This followed the nomination of Heinrich III Parler as Engineer of the Cathedral (appointed from December 1391 until July 1392),°° and it was as a result of his criticisms of the recently accepted project (Stornaloco II), that the general meeting, with 35 Deputati and 14 engineers, was called. At the end of this meeting the new heights for the principal elements of the building were ratified, thus switching from Stornaloco I (28, 42, 56 and 84 braccia) to Stornaloco II (28, 40, 52 and 76 braccia). The passage in the document that confirms Valentini’s hypothesis is the dubium (‘ques- tion’) number five, where it was asked (Appendix II, doc. I):
Table 2. Principal heights of Milan Cathedral according to the Stornaloco I and Stornaloco II schemes compared with those given in modern surveys of the building. Dimensions in braccia in italics are converted from metres, but using decimals for convenience; dimensions in metres in italics are calculated from unannotated drawings. a passage of a well-known document recording the general meeting of 1 May 1392 (Appendix II, doc. I). This followed the nomination of Heinrich III Parler as Engineer of the Cathedral (appointed from December 1391 until July 1392),°° and it was as a result of his criticisms of the recently accepted project (Stornaloco II), that the general meeting, with 35 Deputati and 14 engineers, was called. At the end of this meeting the new heights for the principal elements of the building were ratified, thus switching from Stornaloco I (28, 42, 56 and 84 braccia) to Stornaloco II (28, 40, 52 and 76 braccia). The passage in the document that confirms Valentini’s hypothesis is the dubium (‘ques- tion’) number five, where it was asked (Appendix II, doc. I):
Fig. 10. Antonio di Vincenzo (attributed) or copyist, plan and cross-section of Milan Cathedral (Bologna, Archivio della Fabbriceria di S. Petronio; with permission)
Fig. 10. Antonio di Vincenzo (attributed) or copyist, plan and cross-section of Milan Cathedral (Bologna, Archivio della Fabbriceria di S. Petronio; with permission)
These are supplemented on the verso (Fig. 12) by the following inscription:
These are supplemented on the verso (Fig. 12) by the following inscription:
Fig. 14. Detail of ground-plan of Milan cathedral (from C. Cesariano, Di Lucio Vitruvio Pollione de Architectura, 1521, 14r). The plan, based on an older plan dateable to 1390-94, shows the reinforced piers below the cupola, the doors of the sacristies in their present positions and the sacristies without their vaults
Fig. 14. Detail of ground-plan of Milan cathedral (from C. Cesariano, Di Lucio Vitruvio Pollione de Architectura, 1521, 14r). The plan, based on an older plan dateable to 1390-94, shows the reinforced piers below the cupola, the doors of the sacristies in their present positions and the sacristies without their vaults
Fig. 15. Diagram based on Nava’s copy of Stornaloco’s drawing (after Mella, Elementi di architettura gotica da documenti antichi, 1863)
Fig. 15. Diagram based on Nava’s copy of Stornaloco’s drawing (after Mella, Elementi di architettura gotica da documenti antichi, 1863)
Related topics:
Medieval HistoryHistory of architectureGothic Cathedral constructionTheories of ProportionHistory of MilanArchitectural DrawingDrawing as a tool for design
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