# Singularities of Perspective in Spain: The ‘Other Treatises’ on Perspective and their Importance within the European Context

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## Abstract

Regarding drawing perspective, Spanish tradition has hidden singularities that greatly distinguish it from other European traditions. Spanish tradition, however, simultaneously nurtures ideas from these traditions, such as those of Italy and France, creating multiple variations and novelties. This study presents a brief journey through the lesser-known exempla, which are lesser studied in the scientific community. The study focuses on the ‘angular perspective’ by architect Hernán Ruiz the Younger (el Joven), the innovative construction of figures in perspective by the carpenter Antonio de Torreblanca and the singular contributions to the anonymous manuscript *Artes Exçelençias dela Perspectiba.* All three studies are similar in that they feature novel and heterodox premises within the general theory.

## Keywords

Historical treatise Manuals Perspective Geometric construction Representation of architecture Polyhedra## Introduction: Perspective Manuscripts in the Spanish Tradition

As it is commonly known, the study of perspective is a multidisciplinary approach comprising a number of individual yet interrelated areas of knowledge. Despite essentially being an issue of language and communication, its expression is also a question of ideology. The study of perspective is also, in its purest form, an instrumental issue; it is both geometric and technical which enables its use and application (Gentil Baldrich 2012b).

Accordingly, we cannot speak of a single tradition on perspective in Spain since it has evolved over the centuries. Its grammar has been constructed from different axioms and even fashions. Thus we shall speak of a unique ‘angular perspective’ in Spain, inspired by the scientific teachings of Euclid and crossed with an Italianized vision of perspective that eventually dominated artistic circles and guilds. This was made possible by direct contact with Italian and Flemish artists in the cultural melting pot that was the works of Phillip II’s Escorial palace in Madrid. Also key was the arrival of prestigious treatises, full of innovative and impressive illustrations, such as those of Vignola and Danti (1583) and Sirigatti (1596).

The eventual use of perspective construction based on the intersection of the visual pyramid with the picture plane rather than more abstract methods must also be analyzed within an evolutionary logic. It is the result of the advantages of this particular method in saving time and effort over more confusing and complex methods. The methodological evolutions that followed, from the ‘seconda regola’ by Vignola and Danti, the ‘punctus contingentia’ by Marolois (1633), or the perspectograph, up to the *camera obscura* and finally the photograph, also seek ‘the easiest way’, as previously expressed by Albrecht Dürer.

However, we should bear in mind that, besides painters, representing space on a plane is also an issue for architects, carpenters and engineers (Burucúa 1989-1990-1991). Any builder in a complex production system going beyond medieval guild logic needs a method to define and control objects in space starting from a projection plane. In accordance with this they must measure and define all the geometric parts, create templates, calculate expenses, and finally visualize the work before its construction. In this way the evolution reflected in the perspective treatises reveal the true manuals of graphic science in each historical period including the geometric corpus, and theory of polyhedral and projection issues. These form the basis from which all modern systems of representation have been defined, from Monge (1797) to the arrival of CAD technology.

In Spain there are key examples of this trend, outside of pictorial tradition, which deal with the graphical problems within construction crafts. These manuscripts had a private and professional use but in other cases were conceived as real treatises, even prepared for editing but then never published. Within this group of manuscripts the following have been studied: Ruiz (1560),^{1} Antonio de Torreblanca (1600, 1616–1617),^{2} Salvador Muñoz (1642), Felipe Lázaro de Goiti (1643), Luis Carduchi (1650), Simón García (1681), Fray Andrés de San Miguel (1652), Juan Andrés Ricci de Guevara (1659), and the magnificent, also anonymous manuscript *Artes Exçelençias dela Perspectiba* (Anonymous 1688).^{3}

Three unpublished manuscripts have been selected from this group; they are of particular interest due to their little known contributions to the field of perspective: The manuscripts of Hernán Ruiz the Younger and Antonio de Torreblanca, as well as the anonymous work entitled: *Artes Exçelençias dela Perspectiba*. The reason for this selection is that they share a common ground; the three manuscripts contain postulates that are complementary or divergent from the theories generally accepted in Europe at the time. These new postulates are of great interest as they tackle obscure or rejected geometry practices which have stayed in the dark until now. The fact that conic perspective eventually reached a geometric corpus that was widely accepted did not prevent individual research into lesser studied heterodox issues from appearing. This paper outlines the cases found in the Spanish sphere.

## An Angular Perspective in Sixteenth Century Spain

^{4}which remained constant in all Greek authors dealing with optics such as Geminus, Proclus, and Ptolemy. These axioms were studied and discussed by other scholars in the Middle Ages—Al Kindi, Bacon, Peckham, Alhazen, and Vitelius—until the elaboration of a theory of vision known as

*perspectiva naturalis.*This theory of vision contrasts with the

*perspectiva artificialis*that would later be applied to painting and graphic expression of space. In certain circles this precept was preserved due to the prestige that its ancient origin provided. This was claimed by Juan de Herrera, the greatest Spanish architect of his period, director of architecture of El Escorial and first director of the Academy of Mathematics.

^{5}In his work

*La Institución de la Academia Real de Matemáticas*, Herrera recommends perspective scholars read

*“*the first six books by Euclid, also his eleventh and twelfth book; the perspective and

*specularia*by Euclid and Ptolemy, as well as the legacy of these theories in the works of Alhazen and Vitellion” (Herrera 1584: 14). At this time, Euclid’s

*Elements*had already been translated into Spanish by Zamorano (1576), and precisely one year after the publication of Herrera’s work, another text by Euclid was translated by Ambrosio Ondériz (1585) entitled

*Perspectiva y Especularia*. In this way, in his important work,

*Commentariorum in astrolabium*, Juan de Rojas Sarmiento (1551) illustrates the way to measure buildings using his instrument, which provided him angular readings (Fig. 1). This book was undoubtedly in circulation in Seville, whose university currently conserves three copies. It also awoke the interest of architects such as Juan Bautista de Monegro, who had a copy in his library.

For this very reason, there is a marked preference towards angular perspective in Spain, which is the very theory found in the manuscript written by Hernán Ruiz the Younger (1560) and presented in literature which is not widely disseminated (Gentil Baldrich 1998). His manuscript—exact date unknown but developed during his lifetime—contains a compilation of what may be the concerns of the Renaissance architect, including a crucial part on conical perspective, which is discussed in this paper.

From the point of view of V, the floor projection, we outline the angle spanning segment B-C. With an arch from centre V, we find the springing line B-c’, which is transported to the plane of the square to define segment B-c. Similarly, we find its symmetrical pair A-d, determining the lateral foreshortening of the depth in perspective. In the plane of the section starting from the top, we draw the angle from V’ for segment I-J. With an arch with the circumference around centre V’, we mark the springing line i’-J, which is transported with a rotation around centre J, to its position in the picture plane i-J. In the highest part of the section (Fig. 4) we see segment i-J is shorter than that which would have been produced using Leon Battista Alberti’s procedure of intersection in the visual pyramid with the picture plane.

The auxiliary lines drawn from i-J and c-B, d-A, determine positions i1 and i2 of the perspective. The intersection of segment I-i1 with J-i2 determines point 1, the vanishing point of the set of lines running perpendicular to the plane of the square, which are contained within this upper horizontal plane.

In the same way we find segment h-K, which, using auxiliary lines, defines the perspective points h1 and h2 as well as the vanishing point 2 for the middle horizontal wall section. Likewise, the points of the lower floor of g1 and g2 are determined and vanish at 3. The perpendicular lines in the left-hand vertical wall sections (lower and upper) vanish at 4 and 5 respectively. Points 6 and 7 are where the lines in the right-hand wall sections vanish. We can assume that the joint vanishing of the upper verticals at point 1 is a mistake, as observed in the manuscript (Fig. 2). We propose the correct vanishing points of 5 and 7, according to the applied logic. Finally, we have a system made up of seven vanishing points (one for each wall section), with a rhomboid shape in the centre part.

## The Innovative Graphic Procedure of Perspective Representation by Antonio de Torreblanca

The second case is the analysis of another unsolved question within the artistic perspective: the mandatory bifocal terms of vision versus the single point of view of the conic projection. This subject could be traced back to the origins of perspective codification and arises in the form of procedure when considering a graphic system dominated by two points of view instead of a single one. In Spain, this question is studied by Antonio de Torreblanca in one of his treatises as the starting point for a procedure that he calls *Regla novedosa* (new rule).^{6}

In the analysis of Torreblanca’s works we are inclined to agree with Navarro de Zuvillaga (1989) that his true sources are in fact Serlio (1545), Barbaro (1569), Vignola and Danti (1583), and Sirigatti (1596). These Italian masters were the common referents in Spain during the seventeenth century. Torreblanca primarily uses Vignola’s *seconda regola*; taking advantage of the structural facility permitted by the distance points with respect to the *prima regola*, which corresponds to the *costruzione legittima* by Leon Battista Alberti.

*puncta concursus.*

*Regla*(rule) of perspective representation, discussed in his second treatise:

Rule worthy of merit by its own excellence, and known by few, I dare not say by none as this would be too bold, but still I do not know of anyone who has studied this subject or written about it, and as such, I doubt anyone has mastered it in this very fashion, as the present work is the result of my effort alone.

^{7}(Torreblanca 1616–1617: fols. 59–59v).

## Study of the Novel *Regla* by Antonio de Torreblanca

Tracing in the horizon two opposed points, on the sides of the image, or outside; the base is divided into a number of perfectly equal sections and each point is linked with all those divisions, obtaining a system of two oblique, crosswise pyramids […] From this point it is easy to accomplish a net of squares parallel to the picture plane. (Chastel and Klein 1989: 199).

*finitix plani*or final picture plane is located on the horizon, limiting the number of transversals to half of the orthogonals, as Fig. 7 shows.

According to Chastel and Klein, the ‘bifocal method’ may be improved, for example by superimposing the orthogonals pyramid onto the side pyramid of the bifocal system, as proposed by Viator (1505). In the same way, the procedure of the ‘distance point’ could have been an evolution of the artisans’ bifocal method after being compounded with the procedure proposed by Alberti. Even if there was some crossover between these two methods and the work of the Italian—as it may be now assumed—their long-term interrelation was not confirmed until the sixteenth century (Chastel and Klein 1989: 200).

In Torreblanca’s work this relation between the workshop procedures related to the ‘bifocal’ method and Vignola’s *seconda regola* (Fig. 7) is clear.

*Regla*in the next folio, Torreblanca uses control points

*o, n, k, m,*representing the true length of segments of the crossover with line

*A–B*, hereby applying the ‘distance points’ method. In turn, line

*A–B*defines the picture plane. One can see the same sequence of points labeled with capital letters

*O, N, K, M,*arranged in the upper line which defines the intersection of this other plane with the picture plane. If a series of auxiliary vertical lines were drawn—which Torreblanca didn’t do—the correspondence between the two groups of points could be verified. Furthermore, the rectangle that circumscribes the cross in both cases touches the picture plane with the same letter

*K*. This point will be especially significant in the following procedure.

*A–B*as in the previous example. The upper parallel line defines its corresponding parallel plane (not called ‘the horizon’ as it is not the same as the visual experience horizon). In the context of mathematics the rule previously described in Fig. 8 will still be valid in Fig. 9. The procedure could easily be explained if we consider that the orientation has rotated 33° (Fig. 10) to carry out the procedure previously described in the case of the top cross. In this same illustration we can see that the point of view is not represented, exactly as the bifocal system postulates.

*Regla*Torreblanca aims to generalize a well-known procedure not only to control the perspective of objects contained in planes parallel to the geometrical plane, but also for figures contained in any plane perpendicular to the picture plane (Figs. 11, 12). His rule could be understood as a generalization of the bifocal system based on the distance points. This enables one to arrange perspective projections of figures contained in any plane perpendicular to the picture plane.

## The Unpublished Manuscript *Artes Exçelençias* *dela Perspectiba*

The third case analyzed in this paper is a contribution from one of the four books contained in the anonymous manuscript treatise, *Artes Exçelençias dela Perspectiba*. This is kept in the private Archives at the Casa Medina Sidonia Foundation in Sanlúcar de Barrameda, Cádiz, Spain. Unlike the cases previously mentioned this text neither aims to reformulate angular axioms nor to reconsider ambiguous procedures in the canonical trend.^{8} Its contribution focuses on a particular model of perspective restitution that endows bodies with a geometric dimension starting from just one image (a perspective). Thus, the author accomplished an amazing interpretation of complex illustrations dating from the period.

The study, *Artes Exçelençias dela Perspectiba,* is a unique piece in Spanish treatise tradition that contributes to the evolution and singularity in the development of perspective studies during the seventeenth century (Martín-Pastor 2009). The unfinished manuscript comprises four books; the first two are complete and deal with theoretical and abstract aspects of perspective representation and its practical development in architecture (Martín-Pastor and Granado-Castro 2015). The third book studies the graphic construction of quadratures and architectural *trompe*-*l’œil* creations. Finally, the fourth book, which contains the contribution discussed in this paper, focuses on the analysis of the polyhedra and the problem of restitution.

## The Interpretation of Jamnitzer’s Polyhedra

*Perspectiva Corporium Regularuim*by Jamnitzer (1568) goes unmentioned by the author (Fig. 13). It is precisely this omission, quite rare given the nature of the work, which leads us to think that the author probably studied an anonymous copy of Jamnitzer’s work entitled

*Sintagma*, which was in circulation in that period.

As no one before has written about this subject, or at least not the scholars I have met. Most of the brightest ones do not dare to expound the method of representation for regular solid bodies which provide such curious speculations. Through my work and dedication I have obtained a reward that many others did not achieve because of their neglectfulness, and my happiness is the result of their laziness.

^{9}(Anonymous 1688: fol. 57v).

*Sintagma*is an innovative approach without precedent at the time. The author’s contribution consists in devising a system that firstly allows the deducing of metric or proportional information simply from perspective illustrations. Also it would allow the construction of polyhedra with pedestals, truncations, bevels, cavities, etc., starting from an exploratory system of base proportions (Fig. 14).

This hypothesis is based on the existence of a generating order across all graphic construction. On the one hand there is a strict order in the perspective construction. On the other hand there is an absolute order in the system of proportions or modules that coordinates the formal apparatus of representation specifically to justify the creation of fanciful pedestals under the polyhedra (as observable in the lower part of Figs. 14, 16). Clearly, as in the case of the architectonical orders approached in his previous book, the dimensional control of a form should obey a very specific law. A law that in this case will be codified in the form of a graphic procedure which must be formulated as a relation of points and lines in the projection plane (Figs. 14, 16).

*Sintagma*is a subject that our author does not mention. There is not one explanation in the whole treatise about how such complex tridimensional geometry is metrically deduced. Also, it also remains unexplained how the author can find the specific position of the point of view with respect to the object and the picture plane. Indeed, when comparing the illustrations in the manuscripts with the engravings, it may be observed that the position of the point of view with respect to the object changes greatly (Figs. 13, 14). All these interesting questions are illustrated with a testimony, almost a confession, which is placed in the last chapter of the book: the introduction of certain paper models.

…Through my work and dedication, (…) I have achieved the representation of bodies which I made of paper using geometric rules and precepts with the help of their physical and real constitution.

^{10}(Anonymous 1688: fol. 57v).

In constructing these bodies, prior knowledge of the reference model’s geometry (the regular polyhedron before variations) would solve the metric problem underlying the operation. The coincidence and validation criterion of the restitution of the complex piece would be determined by trial-and-error, starting from the visual experience of the model generated from the original engraving. Notwithstanding, it appears that moving those scaled objects with respect to a station point would also be undeniably useful when creating each illustration in a similar fashion to (though not exact replicas of) the reference engraving.

## Conclusion

The study of these three particular cases from three Spanish manuscripts on perspective aims not to provide definitive answers to define the cultural and scientific background of perspective and the systems of architectural representation. Rather, these examples show testimonies of great value regarding the way of understanding perspective in different variations that, in every period and place, have diverged from the canonical trend. The first two cases provide graphic interpretation to unsolved aspects from outside orthodox perspective theory. In the third case a step forward is taken towards the possibility of the perspective restitution of the image.

Hernán Ruiz’s manuscript on perspective depicts a scientific sphere in sixteenth century Seville that knew and applied the geometric theory of ancient authors. The practice resulting from Euclid’s theorems was considered so prestigious in this circle that it made this Renaissance artist follow them despite the advantages of the more simple postulates of the Italian model proposed by Serlio. The manuscript portrays an interesting case of revision of the angular paradigm in a specific context that led a Renaissance scholar and architect to sketch a particular method of drawing perspective. To follow Euclid’s principles in optics was a rather more complex method to accomplish but was not against the artists’ cultural beliefs.

The manuscript by Antonio de Torreblanca, on the contrary, is a fine example of a perspective treatise and is both well structured and well documented. Notwithstanding, a deeper analysis depicts an artist that, as a good artisan, truly understands the art by means of experience and ingenuity, and is therefore capable of presenting a novel proposal. In his own words: “to my knowledge, no one has studied [this subject] or written it yet. Maybe it could be possible to find other instances, though expressed differently, as the present work is the result of my effort only.” (Torreblanca 1616–1617: fols. 59–59v). This study also revealed the survival of archaic formulae of perspective projections in the circle of artisans surrounding Torreblanca even in the seventeenth century. In this case the formulae were for bifocalism but this also happened with the angular perspective by Hernán Ruiz, which was still being mentioned by Simón García in 1681.

Finally, the fourth book of the anonymous treatise *Artes Exçelençias dela Perspectiba,* portrays the first case of ‘graphic hermeneutics’ in the history of Spain. This entire book focuses on the study and reinterpretation—exclusively from the graphic point of view—of the mysterious illustrations compiled in “*Perspectiva Corporium Regularium*” by Jamnitzer (1568).

To some extent, though largely unexplored, the ancient, heterodox reminiscences displayed here must somehow perdure in other European spheres and would prove an interesting field of research for scholars.

## Footnotes

- 1.
- 2.
- 3.
First study approach to the manuscript: Gentil Baldrich and Martín-Pastor (2006). A more complex research study by Martín-Pastor (2009). There is a facsimile edition of the treaty with analysis and transcriptions by Martín-Pastor et al. (2010). Gentil Baldrich (2012a) provided a chronicle on the eventful whereabouts of the manuscript up to its facsimile reproduction. A study on the architectural representation of the treaty can be found by Martín-Pastor and Granado-Castro (2015), and a first study on the manuscript polyhedra was done by Gentil Baldrich and Martín-Pastor (2015).

- 4.
With respect to perspective, the reference is to Euclid’s 8th theorem: the apparent difference between two equal dimensions approached from unequal distances is determined by the relation of the corresponding visual angles. Time held firmly, and without exception, the assumption that visual dimensions are not determined by the distance between the eye and the object, but rather by the measure of the visual angle (Panofsky 1927: 19).

- 5.
The Royal Academy of Mathematics was an institution established by Phillip II in 1582 on Juan de Herrera’s initiative.

- 6.
The first version,

*Los siete Tratados de la perespectiva pratica*…(c.1600), and a later, more complete version entitled,*Los dos libros de geometría y perespectiva pratica…*(1616–1617). The new rule appears in the second. - 7.
Translated from original quotation: “…digo que no ha venido a mi noticia que de nadie aya sido escripta ni sabida… dudo la sepan de la misma manera aunque diferentemente podrá ser se encuentren los pensamientos porque es puro trabajo mío.” (Torreblanca 1616–1617: fols 59r–59v).

- 8.
- 9.
Translated from original quotation: “Como ninguno de quantos hasta aqui han escrito sobre la materia que tratamos ò por lo menos ninguno de los que hasta aqui he visto siendo la mayor parte de los que mejor se han aplicado no exponen el methodo a representar los cuerpos solidos regulares en que tan curiosas especulaciones se pueden hacer yo he solicitado a costa de mi trauajo y aplicación, la gloria que otros no han obtenido ò por su negligencia y mi buena dicha u por su inuencible pereça…” (Anonymous 1688: fol. 57v).

- 10.
Translated from original quotation: “…yo he solicitado a costa de mi trabajo y aplicación, (…) poniendo en práctica la representación de los cuerpos para que me he ayudado de su real y física constitución habiéndolos hecho y formado de papel con la regla y fundamento geométrico” (Anonymous 1688: fol. 57v).

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