EGA-Aproximacion al analisis sistema diedrico español

8/22/2019 EGA-Aproximacion al analisis sistema diedrico espaol

1/9

Concepto espacialTeniendo en cuenta que la mani-festacin de la representacin grfi-co-geomtrica no es sino expresinde la evolucin de un concepto es-pacial especfico asociado a un de-terminado contexto cientfico, el sus-trato epistemolgico sobre el que seerige la clsica disciplina acadmicapermite caracterizar el orden sintc-tico en que se apoya el modelo pro-yectivo como formulacin orientadaa estructurar el pensamiento espacial

del estudiante (Fig. 2).Proponemos dos hiptesis de parti-da con un claro componente dialcti-co en su enunciacin. En primer lugarse considera que el marco epistemol-gico que sustenta la gramtica del sis-tema didrico espaol prioriza la re-presentacin proposicional sobre laanalgica (Ballesteros Jimnez, 1993)y, en consecuencia, establece determi-nadas mecnicas deductivas a partir delas variables lgicas de la operatividadproyectiva. Dicha doctrina es, en prin-

cipio, ajena al proceso psicolgico deformacin de conceptos espaciales atravs de la construccin de imgenesmentales. En segundo lugar se consi-dera que el direct methodanglosajnsupera los problemas del modelo pro-

IntroduccinFrente a la pedagoga constructivis-ta, el descrdito de la enseanza con-ductista se refiere a obviar los proce-sos mentales del estudiante en la tareade construir su aprendizaje. La mec-nica de restitucin espacial en el siste-ma didrico considera la posibilidad derecuperar la estructura tridimensionalde la configuracin geomtrica repre-sentada. Tal proceso se ha tratado deaplicar igualmente para justificar la ob-tencin de la imagen mental corres-

pondiente (Taibo 1966 [1944], 2). Es-ta investigacin pretende poner demanifiesto que la gramtica del siste-ma didrico espaol, heredera de la cl-sica geometra descriptiva crea unas ex-pectativas formativas de difcilfundamentacin en torno al desarrolloy estructuracin del pensamiento es-pacial del estudiante.

A lo largo de la historia la comu-nicacin grfica, ha desarrollado va-riantes e hbridos descriptivos en fun-cin de las necesidades prcticas de

representacin. Durante el Renaci-miento las representaciones tcnicasfueron utilizadas para describir obje-tivamente la forma tridimensional par-tiendo de la sntesis visual de su con-figuracin geomtrica (Fig. 1).

56 APROXIMACIN AL ANLISIS DEL SISTEMA DIDRICO ESPAOL COMO LENGUAJEVctor Grassa-Miranda, Roberto-Vicente Gimnez Morell

En la percepcin de la forma reside el inicio de la formacin de conceptos. Arnheim (1998 [1986] 40)

Palabras clave: Visualizacin, Diseo tridimensional, Imagen mental, Representacin geomtrica.

Mientras la gramtica del sistemadidrico espaol recurre a la formu-lacin del modelo proyectivo paraestructurar el pensamiento espacialdel estudiante, el direct methodan-glosajn se apoya en la reconstruc-cin de la imagen mental de la con-figuracin geomtrica. Lasuperacin del marco epistemolgi-co que da origen a la clsica geo-metra descriptiva conduce a unprogresivo abandono de la especu-lacin intelectual y su reorientacinhacia el diseo aplicado. El retrai-miento cultural de la poca en elcontexto espaol trae como conse-cuencia, en este caso, que la insti-

tucin educativa permanezca ajena

al avance del conocimiento en lamateria, lo que va a propiciar uncierto grado de academicismo que

todava va a permanecer durante lasegunda mitad del siglo XX.


2/9

yectivo al plantear, desde el inicio, laasimilacin del concepto espacial fa-voreciendo la construccin de una ima-gen mental ntida del diseo tridimen-sional. En tal sentido el direct methodse ocupa de potenciar la orientacin es-pacial del estudiante mediante la ob-tencin de vistas ortogrficas de snte-sis perspectiva para apoyar la cognicinde las relaciones espaciales (Fig. 3).

Para validar tales hiptesis indaga-mos en la metodologa operativa de larepresentacin grfico-geomtrica va-lorando cuatro bloques bibliogrficosen los que hemos clasificado el mate-rial existente. El primero de ellos iden-tifica las publicaciones afines a la dis-ciplina acadmica en el contextoespaol: Taibo (1966 [1944]), Izquier-do (1956) y Rodrguez de Abajo (1958)entre las de mayor difusin (Fig. 4).

tual del contenido y evitar la distor-sin que la traduccin pudiera aadir.La versin completa del estudio formaparte de la tesis doctoral de Grassa-Miranda (2008).

Anlisis vs sntesisEl filsofo y matemtico Ren Des-

cartes (1596-1650), impulsor de la ge-ometra analtica, ya haba consideradola posibilidad de reconstruir la realidaddeductivamente a expensas de la expe-riencia. Su concepcin racionalista delmundo, tuvo una profunda influenciaen el continente europeo con su epicen-tro en Francia. En consistencia con elrazonamiento proposicional, el creadorde la geometra descriptiva GaspardMonge (1746-1818) se propone inter-pretar las relaciones espaciales en basea determinadas variables lgicas que sur-gen de la operatividad proyectiva (Bo-oker, 2001 [1963], 25). La motivacinpor extender la legitimidad de las ope-raciones de anlisis al terreno de la re-presentacin grfica va a ser uno de losprincipios embrionarios de esta formu-lacin (Sakarovitch, 1998, 261) (Mon-ge, 1996 [1803], 90) (Fig. 6).

Como fundador de la cole Poly-technique, la posicin institucional ypoltica de Monge justifica, en buenamedida, el papel privilegiado de la ge-ometra descriptiva para constituirse

En un segundo bloque consideramoslas publicaciones renovadoras sobredidrico directo que, no obstante, si-guen sin emanciparse del concepto es-pacial propio del modelo proyectivo.El principal representante de esta co-rriente es Snchez (1997 [1993]). Entercer lugar analizamos las traduccio-nes al espaol del direct methodque,con ms de tres dcadas de retraso, lle-gan al contexto espaol ofreciendo unaversin descontextualizada respecto asus fundamentos originales: Warnerand Mcneary (1964 [1934]), Rowe andMcfarland (1967 [1939]) y Wellman(1964 [1948]) entre los autores ms re-levantes. Finalmente se contemplan laspublicaciones originales del direct me-thod, prcticamente sin presencia enlas bibliotecas espaolas: Millar (1922[1913]), Hood (1926), Warner andMcneary (1934), Rowe and Mcfarland(1946 [1939]) y Wellman (1948), co-mo autores fundadores de esta corrientealternativa (Fig. 5).

El anlisis bibliogrfico de los blo-ques reseados se refiere, de maneraparticular, a la secuencia de conteni-dos y el equilibrio terico-prctico, sinobviar el inters que tienen las cons-trucciones geomtricas como expre-sin de un concepto espacial aplicado.La adopcin del idioma original en al-gunos conceptos y citas, responde a laintencin de mantener la identidad tex-

1. Estudio de una cabeza mediante varias vistasasociadas. Albrecht Drer.


3/9

tructura del diseo tridimensional enbase a las mecnicas deductivas de unaorganizacin racional independientede la experiencia (Grassa-Miranda,2009). Si el significado de la realidadse establece a expensas de las percep-ciones del sujeto, no se hace necesarioverificar los conceptos mediante ob-servacin emprica y, por tanto las im-

genes mentales resultan intiles paraasimilar las relaciones proyectivas. Es-ta situacin introduce, asimismo, unadiscontinuidad respecto al carcteranalgico que haba acompaado a larepresentacin de formas desde el Re-nacimiento, y el espritu de observa-cin de la naturaleza yacente que ha-ba dado preponderancia a la imagenvisual (Fig. 7).

ge la axonometra ortogonal (). Por otraparte la perspectiva cnica era en la escuelade Monge mera aplicacin de la geome-tra descriptiva, ms para demostrar la ca-pacidad operativa del mtodo de Mongeque como reproduccin del viejo procedi-miento de Alberti. De la Gourniere sien-te nostalgia de la riqueza de procedimien-tos grficos de los gremios y la variedad ymatices en los modos de representar, cor-tada por la imposicin de una lengua nor-

malizada. Su crtica se completa con la rei-vindicacin de un terreno propio para laperspectiva cnica y la caballera. (Monge,1996 [1803], 91).

El estatus de sistema otorgado alos procedimientos perspectivos, ten-dr la contrapartida de someter la cog-nicin espacial a las variables lgicasde la operatividad proyectiva. En talsentido, se trata de interpretar la es-

en una de las disciplinas acadmicasdel modelo napolenico de formacinpolitcnica y su posterior difusin (Bo-yer, 1982 [1968], 598), contando ade-ms con la enorme influencia que Fran-cia ejerca en gran parte del continenteeuropeo. La normativizacin que pro-mueve el modelo proyectivo da lugar,durante la primera mitad del siglo XIX,

a un proceso de reinterpretacin de tc-nicas que determinar la codificacinde los sistemas de representacin:

Ya hemos mencionado la supresin en lacole Polytechnique de las perspectivas ca-balleras de las lminas de De la Rue. A pe-sar de que este sistema de representacinera genuinamente francs, no se le prestatencin en Francia hasta los escritos deTheodore Olivier. An menos habran deestudiar los primeros discpulos de Mon-

58

2

3 4

2. Alfabeto del plano (Gutirrez Vzquez; IzquierdoAsensi; Navarro de Zuvillaga; Placencia Valero;1984, p. 121).3. Proceso de visualizacin/representacin de obje-

tos tridimensionales (Bertoline, 1997, 241).4. Lmina 2 de la gomtrie descriptive(Monge,1996 [1803]).

5. Direct method. Vistas auxiliares. Giesecke(1979, 278).


4/9

El origen del direct methodQu posibilidades existen para fun-

damentar la capacidad de estructurarel pensamiento espacial a travs del sis-tema didrico espaol? Hace aproxi-madamente un siglo, en la AnnualConference of the Society for the Pro-motion of Engineering Education, elprofesor French (1976, [1913], 22) ex-

pona las diferencias entre dos posiblesmodalidades de representacin grfi-co-geomtrica:

The methods now in use, with alltheir variations, may be divided intotwo general classes:

1. Those which begin with the theory ofthe point, line and plane, and progress tothe solid.2. Those which begin with the solid, andafterwards take up the analysis of linesand surfaces.()In a subject which depends wholly on theclearness of perception, and whose va-lue is entirely lost if the mental picture isconfused, the possibility of this conditionis most unfortunate.

Mientras la primera opcin, here-dada de la formulacin proyectiva seocupa de teorizar en torno a las abs-tracciones del punto, la recta y el pla-no, la segunda se inicia en la visuali-zacin del diseo tridimensional como

origen para interpretar su estructurageomtrica. Esta propuesta considerael papel fundamental de la imagenmental en la cognicin de las relacio-nes espaciales, aspecto que ser deter-minante para la gnesis posterior deldirect method(Fig. 8).

Adam V. Millar (1873-1960) seconsidera el primer autor en realizaruna propuesta alternativa a la for-

mulacin del modelo proyectivo(Gonzlez et al. 1977, 3), pero es Ge-orge Jssen Hood (1877-1965)quien, pocos aos despus, realiza unamplio desarrollo de todas las con-secuencias del direct method. Estametodologa favorece la interaccinsujeto-objeto mediante la construc-cin de vistas ortogrficas de snte-sis perspectiva con el fin de estimu-lar la orientacin espacial delestudiante. Cada problema de dise-o tridimensional dispone de un pun-to de vista estratgico desde el cualpoder abordar su solucin. Por talmotivo el direct methodse ocupa,desde un principio, de ejercitar al es-tudiante en el modo de obtener vis-tas auxiliares y construir una imagenmental ntida del diseo tridimen-sional (Fig. 9).

En este sentido, los desarrollos deldirect methodmantienen un alto gra-do de identidad entre representaciny visualizacin tal y como se haba en-tendido en la tradicin renacentista:

When the engineer draws or reads a view,he visualizes the views as representing thesolid three dimensional object. It is a mis-take to think that the object is projectedon a plane, or regard the view as flat. (Ho-

od 1946 [1926], 19)El direct methodpropone tratar sin

intermediarios con la configuracin es-pacial, lo que significar derogar elaparato deductivo-cartesiano de la ge-ometra descriptiva clsica. En pala-bras de Hood el direct method tratacon el objeto mismo y requiere de una

actitud mental diferente (Hood et al.1979 [1969], 445).

Si la visualizacin perspectiva re-sulta un soporte cognitivo eficaz porqu no integrarla dentro de la repre-

sentacin ortogrfica? La combinacinde diferentes vistas ortogrficas enperspectiva del diseo tridimensionalconstituye una de las caractersticasms destacada del direct methodcuyofin especfico es facilitar la compren-sin y el anlisis de la configuracingeomtrica (Fig. 10).

5


5/9

cuestionado en base a una insuficien-te fundamentacin del proceso cons-tructivo de la imagen mental de la con-figuracin geomtrica que propicia eldesarrollo de una secuencia metodo-lgica alejada de la praxis. La co-rriente anglosajona del direct methodsoluciona este problema al plantear,desde el inicio de su secuencia didc-

tica la asimilacin del concepto espa-cial favoreciendo la construccin deuna imagen mental ntida del diseotridimensional. La superacin del mar-co epistemolgico que da origen a laclsica geometra descriptiva condu-ce a un progresivo abandono de la es-peculacin intelectual, y su reorienta-cin hacia el diseo aplicado. Talrenovacin se desarrolla en base a unametodologa alternativa, consistenteen obtener vistas ortogrficas de sn-tesis perspectiva como modo de po-tenciar la interaccin sujeto-objeto.Ajena al avance del conocimiento,eneste caso, el retraimiento cultural dela institucin educativa espaola de lapoca, va a propiciar el academicismoen una la disciplina que, fiel a los pre-ceptos de la tradicin clsica, todavase prolongar durante la segunda mi-tad del siglo XX.

60

6

8

7

ConclusinEl estatus de lenguaje que ostenta

la gramtica del sistema didrico es-paol como organizacin racionalorientada a estructurar el pensamien-to espacial del estudiante podra ser


6/9

Referencias bibliogrficas

ARNHEIM, R. 1998 [1986]. El pensamiento visual. Paids,Barcelona. Visual Thinking, 1969.

BALLESTEROS JIMNEZ, S. 1993, Representaciones anal-gicas en percepcin y memoria: imgenes, transforma-

ciones mentales y representaciones estructurales.

http://www.psicothema.com/pdf/858.pdf (octubre 2009)

BERTOLINE G. R. ETAL. 1997, Technical Graphics Communi-cation. En castellano: Dibujo en Ingeniera y Comunica-cin Grfica. McGraw-Hill, Mxico.

BOOKER, P. J. 2001 [1963]. Una Historia del Dibujo en In-geniera. Centro Asociado de la UNED, Jan. A Historyof Engineering Drawing. Chattoo and Windus. Publicadaposteriormente por Northgate Publishing, 1979.

BOYER, C. B. 1982 [1968]. Historia de la matemtica. Alian-za editorial. Madrid. A History of Mathematics. J Wiley

& Sons, Inc. FRENCH, T. E. 1976 (1913). The educational side of en-

gineering drawing, The Engineering Design GraphicsJournal, 40 (3):32-35.

GIESECKE, F. E. ETAL. 1979. Dibujo para ingeniera. Mxi-co, Interamericana. [1942] Technical drawing, includingaeronautical drafting. The Macmillan Company. New York.

GIMNEZ MORELL, R. V. 1988. Espacio, visin y representa-cin en el Dibujo y en la pintura del sigloXX. Servicio dePublicaciones Universidad Politcnica. Valencia.

GONZLEZ GARCA, V. LPEZ POZA, R. Y NIETO OATE 1977. Sis-temas de Representacin. Texgraf. Valladolid.

GRASSA-MIRANDA, V. 2009. Cognicin espacial en la re-presentacin grfico-geomtrica. Arquiteturarevista, v.5 n 1. http://www.arquiteturarevista.unisinos.br/

GRASSA-MIRANDA, V. 2008. Lectura y evaluacin del espa-cio tridimensional en la representacin grfico-geomtri-

ca. Tesis doctoral dirigida por Dr. Roberto Gimnez Morell.Universidad Politcnica de Valencia. 28 de octubre de 2008.

GUTIRREZ VZQUEZ, NGEL; IZQUIERDO ASENSI, FERNANDO; NA-VARRO DE ZUVILLAGA, JAVIER Y PLACENCIA VALERO, JOB. 1984,Dibujo tcnico. Editorial Anaya. Madrid

HOOD, GEORGE J. 1946 [1926]. Geometry of engineeringdrawing: Descriptive Geometry by the Direct Method.

McGraw-Hill, New York.

HOOD, G. J.; A. S. PALMERLEE y CH. J. BAER, 1979 [1969]. Ge-ometry of engineering drawing. Huntington, New York,Robert E. Krieger Publishing Company.

IZQUIERDO ASENSI, F. 1956. Apuntes de Geometra Des-criptiva. Patronato de Hurfanos de Oficiales del Ejr-cito. Madrid.

MILLAR, A. V.; EDWARD S. MACLIN. 1922 [1913]. Descripti-ve Geometry. New York [etc.] McGraw-Hill book company.La edicin manejada es de 1922.

MONGE, G. 1996 [1803]. Geometra Descriptiva. Facsmildel original (1996). Madrid, Imprenta Real. Ed. Colegio

de Ingenieros de Caminos, Canales y Puertos de Ma-drid. (Prembulo de Jos Mara Gentil Baldrich y Enri-que Rabasa Daz).

RODRGUEZ DE ABAJO, F. J. 1957. Dibujo geomtrico. Mar-fil. Alcoy.

ROWE, CH. E. Y J. D. MCFARLAND. 1946 [1939]. EngineeringDescriptive Geometry: The Direct Method for students.New York, Van Nostrand Company, El ejemplar maneja-do es una reimpresin de 1946. En castellano (1967)Ge-ometra Descriptiva. Continental. Mxico.

SAKAROVITCH, J. 1998, Epures darchitecture, de la coupedes pierres la G omtrie Descriptive, XVI-XIX sicles,Basel, Birkhuser.

SNCHEZ GALLEGO, J. A. 1997 [1993]. Geometra Descripti-va: sistemas de proyeccin cilndrica. UPC. Barcelona. La

edicin manejada es de 1997 TAIBO, . 1966 [1944]. Geometra Descriptiva y sus apli-

caciones. Tomo I, Editorial Tebar Flores, Madrid. La edi-cin manejada es de 1966 (2 edicin).

WARNER, F. M. and MCNEARY, 1934. Applied DescriptiveGeometry with Drafting-room problems. McGraw-Hill.New York London.

WELLMAN, L. 1948. Technical Descriptive Geometry. McGraw-Hill. New York. En castellano Compendio de Geometra Des-criptiva para tcnicos. Reverte. Mxico, 1964.

9 10

6. Caricatura de Monge por uno de sus alumnos ha-cia 1802 (Sakarovitch, 1998, 246).7. Estudio arquitectnico para La Adoracin de losReyes Magos, Leonardo da Vinci.8. Direct method. Dinmica para representar/visuali-zar (Hood 1946, 22).

9. Direct method. Representacin de un elemento a tra-vs de diversas vistas auxiliares (Warner, 1934, p. 14).10. Direct method. Vistas auxiliares diversas de unapirmide (Hood, 1946, 25).


7/9

32

drawings of the National Library with those preserved atthe Palace.6/ Antonio PONZ, Viage de Espaa, VI tome, third edition,Viuda de Ibarra (facsimile edition, Madrid, 1972), pp. 89 to91; to accommodate the scale model a box was fitted on thewall of the War Museum, which overlooks the Palaceal-though it was later moved to the workshop under the arch,which communicates with the Garden of the Royal Pharma-cy. Gaspar M. de JOVELLANOS also mentioned this scale

model in his Elogio a Ventura Rodrguez ledo en la real So-ciedad Econmica de Madrid, Viuda de Ibarra, Madrid,1790. Both Pascual MADOZ, Madrid..., 1848, pp. 248 and 333,and ngel FERNNDEZ DE LOS ROS, Gua de Madrid...,

1876, p. 230, 484 and 485, recorded the same dimensionsregistered by Ponz, but Madoz says that the model was keptin the Topographical Cabinet located at the Buen Retiro andFernndez de los Ros informs us of its removal to the Muse-um of Engineers. The new scale model was made for the ex-hibition about Philip V before mentioned.7/ The comparative study of the different projects made forthe new Royal Palace forms part of the graphic documenta-tion gathered by Javier Ortega Vidal, Aitor Goitia Cruz, n-

gel Martnez Daz and Mara Jos Muoz de Pablo for theabove mentioned exhibition El arte en la Corte de Felipe V.8/ Gianfranco GRITELLA, Juvarra. LArquitectura, Modena,1992, t. II, p. 439.

serve the existing houses and small factories whenhe drafted the Preliminary Plan for the Enlargementof Madrid in 1860.If Juvarras Palace had been built in any of the men-tioned places, the growth of Madrid would have hadother features. The nobility with their palaces andgardens and the institutional and administrative build-ings would have occupied this area instead of thedunghills, shanty towns, cemeteries and humbledwellings which formed the suburb originally. Thebuilding up of the Palace would have altered the spa-tial distribution of the social classes and the location

of institutions and services in the city. The urban gridand the typology of the buildings would also havebeen affected.12. A hypothesis about the location of the RoyalPalace designed by Juvarra for the town of Madridabout 1740, superimposed over the present urbangrid, by M. J. Muoz de Pablo, 2008.When we superimpose the projected Palace and thetown existent in the second third of the eighteenth cen-tury over the present urban grid, we can see how someof the roads which were later transformed into prom-enades and later on into the streets which structurethe city, would have been erased by the palace com-plex. This also enables us to imagine a different city,where the structure of the streets would be subject tothe laws imposed by the Palace, and the dimensionsof the blocks and plots would be greater than the ex-

isting ones, a town designed to accommodate the rep-resentative buildings and the sumptuous houses of thearistocracy instead of the constructed town, destinedto house the working quarters. San Bernardinos asy-lum, situated where is nowadays the rectory of theComplutense University, would not have been installedin the ancient convent of the same name, because itwould have been razed to the ground by the worksof the Palace. The Ciudad Universitaria would havehad to find another place to be settled, because itwould have been occupied by the gardens of thePalace. The present square of Moncloa and the wayout of Madrid through La Corua road would have hada completely different shape However, the buildingof Juvarras Royal Palace in the Heights of SanBernardino, although possible, is only a dream; the ac-tual development of the events followed anothercourse which led to the present city.Nevertheless, the drawing up of different plans hashelped us to learn about forgotten ideas from the pastand to speculate about a different present, providingnew images to the rich collection we already haveabout that slipped opportunity. These drawings showthe most forgotten aspect of this subject: the plot ofthe Palace and its relationship with the city. Maybethe knowledge of the past facts and their would-beconsequences will allow us to foresee more clearlya future which is always to come.

ity and the municipal authorities, both in their dailyactivities as administrators and representatives ofpower and in public festivities and ceremonies. Norwere the surrounding establishments, the TapestryManufacture and the Snow Pits, suitable neighboursfor a royal residence. In the second half of the eigh-teenth century, Antonio Carlos of Bourbon projectedfor these lands royal stables which did not come tobe built. At the end of that century a Bleach Filtra-tions Factory was established there. This was also aroyal manufacture, as much as the tapestry one.The two supposed locations, at both sides of San

Bernardinos road, besides their good communica-tion wi th the Royal Residences, would have main-tained a prominent position over the Manzanares riv-er and would have enjoyed the same panoramicviews onto the Casa de Campo as they had enjoyedfrom the Alczar. Because of their being in raisedgrounds, they would have had also better views on-to the town and to the north they would have enjoyedample views onto the Sierra de Guadarrama. Eventhough, it seems more practical to have located thepalace to the west of the promenade, between theroad and the river where the advantages above men-tioned are stressed. There, the smoother relief is tobe found at the eastern margin of the promenade,in front of the Seminary of Noblemen and near theBodyguards barracks.Any one of the places considered could have housed

the new palace, but the two latter are the most like-ly according to the textual description: a place situ-ated in San Bernardino Heights. Finally, we can situ-ate Juvarras palace in a last place which alsoobserves this requisite: in the axis of San Bernardi-nos road and a little further away from the enclosingwall, in the lands which formed the large estate of LaFlorida. This situation seems the most convenient be-cause it joins together the advantages of the previ-ous ones and gives room to the spreading out of thegardens looking onto the north and east proposed byJuvarra in his first outlines, as much as to the lay-ing out of the square before the main faade. The ax-ial composition of this square, the palace and the gar-dens would be strengthened by San Bernardinospromenade, thus avoiding the ambiguity of the ac-cesses typical of the situation tangential to the road.The Palace would be the crowning element of one ofthe main streets in the city.All the sitings we have considered for the buildingwould have changed the shape of Madrid and influ-enced its growth. In the nineteenth century, the ur-ban grid jumped over the enclosing wall. The firstcentre of population to be settled in the outskirts ofthe city was the suburb of Chamber, a humble quar-ter situated to the north of the city between the gatesof Fuencarral and Recoletos where Carlos Mara deCastro situated the working quarter in order to pre-

APPROACH TO THE ANALYSIS OF THESPANISH SISTEMA DIDRICO AS ALANGUAGE

by Vctor Grassa-MirandaRoberto-Vicente Gimnez Morell

The perception of shape is the beginning of conceptuali-sation.Arnheim (1998 [1986] 40)

Abstract

The grammar or guiding principles of the Spanish sis-tema didricouse the projective schema of a mod-el to help structure a students spatial thinking, whilethe Anglo-Saxon direct methodrelies on the recon-struction of a mental image of the geometric con-figuration. The epistemological framework that cre-ated classical descriptive geometry has beensuperseded and this has lead to a progressive aban-donment of intellectual speculation and a reorienta-tion towards applied design. Spains previous culturalisolation meant that educational institutions fell be-hind advances in the field of drawing and a degreeof scholasticism took hold until the second half of thetwentieth century.

Keywords: visualisation, three-dimensional design,mental image, geometric drawing.

IntroductionFrom the viewpoint of constructivist teaching theo-ry, the discredited behaviourist approach ignores themental processes of students as they learn. The me-chanics of restitutionin the Spanish sistema didri-coconsider the possibility of recovering the three-dimensional structure of the geometry represented.Such a process has also been applied to justify ob-taining the corresponding mental image (Taibo 1966[1944] 2). This paper aims to show that the grammaror guiding principles of the sistema didrico, inherit-ed from classical descriptive geometry, create learn-ing expectations that are difficult to anchor with re-spect to the development and structuring of astudents spatial thinking.Throughout the history of graphic communication,various descriptive variations and hybrids have been

developed according to practical drawing needs. Dur-ing the Renaissance, drawing techniques were usedto obj ectively describe three-dimensional shapesbased on the visual synthesis of an objects geome-try (Fig. 1).

Spatial conceptA graphic-geometric drawing is simply an expressionof the evolution of a specific spatial concept associ-ated with a given scientific context. The epistemo-logical platform on which the classical academic dis-


8/9

The origin of the direct methodIs it possible to sustain the ability to structure spatialthinking using the Spanish sistema didrico? Near-ly a century ago, Professor French at the Annual Con-ference of the Society for the Promotion of Engi-neering Education (1976, [1913], 22) outlined thedifferences between two forms of graphic-geomet-ric drawing:The methods now in use, with all their variations, maybe divided into two general classes:1. Those which begin with the theory of the point, lineand plane, and progress to the solid.

2. Those which begin with the solid, and afterwardstake up the analysis of lines and surfaces ()In a subject which depends wholly on the clearnessof perception, and whose value is entirely lost if themental picture is confused, the possibility of this con-dition is most unfortunate.The first option follows from the traditional projec-tive schema and focuses on theorizing about the ab-stractions of point, line, and plane. The second op-tion start s with the three-dime nsion al designvisualization as a source for interpreting geometricstructure. This proposal considers the fundamentalrole of mental imagery in the cognition of spatial re-lations, an aspect that was crucial in the subsequentdevelopment of the direct method (Fig. 8).Adam V. Millar (1873-1960) is considered the first au-thor to make an alternative proposal to the projective

model schema (Gonzalez et al. 1977, 3). A few yearslater George Jssen Hood (1877-1965) made a com-prehensive presentation of the results of applying thedirect method. This methodology favoured subject-object interaction through the construction of syn-thetic or thographic perspectives in order to stimu-late the spatial orientation of students. Eachthree-dimensional design problem has a strategicstandpoint from which to approach the solution. Thisis why the direct method trains the student from theoutset in techniques for producing auxiliary viewsand constructing a clear mental image of a three-di-mensional design (Fig. 9).Developments of the direct method maintain close-ness between drawing and visualization that was un-derstood in the Renaissance tradition:When the engineer draws or reads a view, he visu-alizes the views as representing the solid three-di-mensional object. It is a mistake to think that the ob-ject is projected on a plane, or regard the view as flat.(Hood 1946 [1926], 19).The direct method attempts to deal directly with thespatial configuration and this means repudiating thedeductive Cartesian apparatus of classical descrip-tive geometry. In the words of Hood the direct methoddeals with the object itself and requires a differentmental attitude (Hood et al. 1979 [1969], 445).If the viewing perspective is an effective cognitive sup-

sidered the possibility of reconstructing reality de-ductively from experience. His rationalist worldviewhad a profound influence on Europe and especiallyin France. Consistent with propositional reasoning,the founder of descr iptive geometry, Gaspard Mon-ge (1746-1818), sought to interpret spatial relation-ships based on certain logical variables arising fromprojective operationality(Booker, 2001 [1963] 25). Themotivation to extend the legitimacy of such an analy-sis to the field of graphic drawing was one of the ear-ly embryonic stages of this schema (Sakarovitch,1998, 261) (Monge, 1996 [1803], 90) (Fig. 6).

Monge founded the cole Polytechnique and his in-stitutional and political position helped explain theprivileged role given to descriptive geometry as oneof the academic disciplines of the Napoleonic mod-el of technical university education. The spread ofhis ideas was further helped by the enormous influ-ence that France exercised over much of Europe(Boyer, 1982 [1968], 598). During the first half of thenineteenth century, the standardization resulting fromthe projective model cause d a re interpretation oftechniques for the description of drawing systems:We have already mentioned the removal from thecole Polytechnique of the gentlemanly perspectivesproduced by De la Rue. Although this system of rep-resentation was genuinely French, it was ignored inFrance until Theodore Olivier drew attention to thesystem in his writings. Even fewer have studied the

early disciples of the axonometric orthogonal Mon-ge (...). Moreover, in the school of Monge, conicalperspective was a mere application of descriptivegeometry and was intended as a demonstration ofthe operational capability of Monges method ratherthan as a reproduction of the old procedure of Alberti.De la Gourniere remarked on the imposition of a stan-dardised graphic language and expressed nostalgiafor the earlier wealth of graphical procedures pro-duced by the guilds and the variety and nuances for-merly found in graphic drawing. His critique includesa demand for a legitimate place for the conical andcavalier perspectives. (Monge, 1996 [1803], 91)The status of system given to perspective proce-dures meant submitting spatial cognition to the log-ical variables of projective operationality. The aimwas to interpret the three-dimensional design usingdeductive mechanics in a rational organization thatwas independent of experience (Grass-Miranda,2009). If reality is established at the expense of sub-jective perceptions, it is not necessary to verify theconcepts by empirical observation, and thereforemental images are useless for assimilating projec-tive relations. This introduces a discontinuity with re-gard to the analogue character that has accompa-nied drawing since the Renaissance, and the spiritof the observation of nature that gave prominence tothe visual image (Fig. 7).

cipline stands enables us to characterize the syn-tactic order supporting the projective model as a de-sign-oriented schema that aims to structure the spa-tial thinking of students (Fig. 2).We propose two hypotheses with a clear dialecticalcomponent. Firstly, it is considered that the episte-mological framework that sustains the guiding prin-ciples of the Spanish sistema didricoprioritizespropositional representation over the analogue(Ballesteros Jimenez, 1993); and so establishes cer-tain deductive mechanisms from the logical variablesarising from the projective operatinality. This doctrinelies outside the psychological process of forming spa-tial concepts through the construction of mental im-ages. Secondly, it is considered that the Anglo-Sax-on direct method solves the problems of the projectivemodel by considering, from the beginning, the as-similation of spatial concepts and so favouring theconstruction of a clear mental image of three-di-mensional design. The direct method enhances thespatial orientation of the student by producing ortho-graphic views with synthetic perspectives that facil-itate an understanding of spatial relationships (Fig. 3).To validate these hypotheses we will explore theoperational methodology of graphical-geometricdrawing by dividing the existing bibliography intofour sections.The first section includes the publications that discussthe academic discipline in Spain: Taibo (1966 [1944]),Izquierdo (1956), and Rodriguez de Abajo (1958) being

among the most widely distributed (Fig. 4).In a second section, we consider publications aboutthe sistema didricothat offer a revisionist view, butstill remain tied to the spatial concept of the projec-tive model. The main representative of this trend isSnchez (1997 [1993]). Thirdly, we analyze the Span-ish translations of books about the direct method.These books were published in Spain after a delayof 30 years and offered a de-contextualised versionof the original fundamental principles: Warner andMcNeary (1964 [1934]), Rowe and McFarland (1967[1939]) and Wellman (1964 [1948]) are among the mostrelevant authors. Finally, the original publications practically inexistent in Spanish libraries about thedirect method are considered: Millar (1922 [1913]),Hood (1926), Warner and McNeary (1934), Rowe andMcFarland (1946 [1939]) and Wellman (1948), were

the founding authors of this alternative trend (Fig. 5).The literature review of these sections refers, in par-ticular, to the content and the balance between the-ory and practice, without forgetting the perspectiveof geometric constructions as a manner of express-ing applied spatial concepts. The full version is partof the doctoral thesis of Grass-Miranda (2008).

Analysis versus synthesisThe philosopher and mathematician Ren Descartes(1596-1650) pioneered analytical geometry and con-


9/9

GONZLEZ GARCA, V. LPEZ POZA, R. Y NIETO OATE 1977. Sis-temas de Representacin. Texgraf. Valladolid.GRASSA-MIRANDA, V. 2009. Cognicin espacial en la repre-sentacin grfico-geomtrica. Arquiteturarevista, v. 5 n 1.http://www.arquiteturarevista.unisinos.br/GRASSA-MIRANDA, V. 2008. Lectura y evaluacin del espaciotridimensional en la representacin grfico-geomtrica.Doctoral thesis directed by Dr. Roberto Gimnez Morell.Universidad Politcnica de Valencia. 28 de octubre de 2008.GUTIRREZ VZQUEZ, NGEL; IZQUIERDO ASENSI, FERNANDO; NAVAR-RODE ZUVILLAGA, JAVIER Y PLACENCIA VALERO, JOB. 1984, Dibujotcnico. Editorial Anaya. MadridHOOD, GEORGE J. 1946 [1926]. Geometry of Engineering Draw-ing: Descriptive Geometry by the Direct Method. McGraw-

Hill, New York.HOOD, G. J.; A. S. PALMERLEE y CH. J. BAER, 1979 [1969]. Geom-etry of Engineering Drawing. Huntington, New York, RobertE. Krieger Publishing Company.IZQUIERDO ASENSI, F. 1956. Apuntes de Geometra Descriptiva.Patronato de Hurfanos de Oficiales del Ejrcito. Madrid.MILLAR, A. V.; EDWARD S. MACLIN. 1922 [1913]. DescriptiveGeometry. New York [etc.] McGraw-Hill. 1922 edition.MONGE, G. 1996 [1803]. Geometra Descriptiva. Copy of origi-nal (1996). Madrid, Imprenta Real. Ed. Colegio de Ingenierosde Caminos, Canales y Puertos de Madrid. (Introduction byJos Mara Gentil Baldrich and Enrique Rabasa Daz)RODRGUEZDE ABAJO, F. J. 1957. Dibujo geomtrico. Marfil. Al-coy.ROWE, CH. E. Y J. D. MCFARLAND. 1946 [1939]. Engineering De-scriptive Geometry: The Direct Method for Students. NewYork, Van Nostrand Company, 1946 reprint. In Spanish (1967)Geometra Descriptiva. Continental. Mexico.SAKAROVITCH, J.1998, Epures darchitecture, de la coupe despierres la Gomtrie Descriptive, XVI-XIX sicles, Basel,

Birkhuser.SNCHEZ GALLEGO, J. A. 1997 [1993]. Geometra Descriptiva:sistemas de proyeccin cilndrica. UPC. Barcelona. 1997edition.TAIBO, .1966 [1944]. Geometra Descriptiva y sus aplica-ciones. Tomo I, Editorial Tebar Flores, Madrid. 1966 edition(2nd edicin).WARNER, F. M. and MCNEARY, 1934. Applied DescriptiveGeometry with Drafting-room Problems. McGraw-Hill. NewYork London.WELLMAN, L. 1948. Technical Descriptive Geometry. Mc-Graw-Hill. New York. In Spanish. Compendio de GeometraDescriptiva para tcnicos. Reverte. Mexico, 1964.

FIGURESFig. 1. Study of a head using multiple views. Albrecht Drer.Fig. 2. Plane alphabet (Gutierrez Vazquez; Izquierdo Asensi;Zuvillaga Navarro; Placencia Valero, 1984, p.121).Fig. 3. Visualization and drawing process of three-dimensionalobjects (Bertolino, 1997, 241).

Plate 2 from the Gomtrie Descriptive (Monge, 1996 [1803]).Fig. 5. Direct method. Auxiliary views by Giesecke (1979, 278).Fig. 6. Caricature of Monge drawn by one of his students inabout 1802 (Sakarovitch, 1998, 246).Fig. 7. Architectural sketch for the Adoration of the MagibyLeonardo da Vinci.Fig. 8. Direct method. The dynamics of representation and vi-sualisation (Hood 1946, 22).Fig. 9. Direct method. Drawing of an element using variousauxiliary views (Warner, 1934, p. 14).Fig. 10. Direct method. Various auxiliary views of a pyramid(Hood, 1946, 25).

port then why not integrate the perspective with or-thographic drawing? Combining different orthographicperspectives of three-dimensional design is one of themost prominent features of the direct method the aimbeing to facilitate an understanding and analysis ofgeometrical configuration (Fig. 10).

ConclusionThe status of the grammar and language that ex-presses the guiding principles of the Spanish sistemadidricoas a rational organization aimed at structur-ing student spatial thinking can be challenged on thebasis that it offers an insufficient foundation in the

process of constructing a mental image of the geo-metrical configuration and this prompts the devel-opment of a methodological approach that is distantfrom accepted practice. Secondly, it is considered thatthe Anglo-Saxon direct method supersedes the prob-lems of the projective model by considering, from thebeginning, the assimilation of spatial concepts andso favours the construction of a clear mental image ofthree-dimensional design. Superseding the episte-mological framework that created classic descriptivegeometry leads to a progressive abandonment of in-tellectual speculation and a reorientation towards ap-plied design. This renovation was based on an alter-native methodology that sought synthetic orthographicperspectives as a manner of enhancing the interac-tion between subject and object. As Spain had cul-turally withdrawn from the modern world, the nations

educational establishment remained isolated fromthese advances in knowledge. This isolation led to ascholasticism that remained loyal to the classical tra-dition until the second half of the twentieth century.

REFERENCESARNHEIM, R. 1998 [1986]. El pensamiento visual. Paids,Barcelona. Visual Thinking, 1969.BALLESTEROS JIMNEZ, S.1993, Representaciones analgicasen percepcin y memoria: imgenes, transformacionesmentales y representaciones estructurales. http://www.psi-cothema.com/pdf/858.pdf (October 2009)BERTOLINE G. R. ETAL. 1997, Technical Graphics Communica-tion. In Spanish: Dibujo en Ingeniera y Comunicacin Grfi-ca. McGraw-Hill, Mexico.BOOKER, P. J. 2001 [1963]. Una Historia del Dibujo en Inge-niera. Centro Asociado de la UNED, Jan. A History of En-gineering Drawing. Chattoo and Windus. Later publishedby Northgate Publishing, 1979.

BOYER, C. B.1982 [1968]. Historia de la matemtica. Alianzaeditorial. Madrid. A History of Mathematics. J Wiley &Sons, Inc.FRENCH, T. E. 1976 (1913). The educational side of engineer-ing drawing, The Engineering Design Graphics Journal, 40(3):32-35.GIESECKE, F. E. ETAL. 1979. Dibujo para ingeniera. Mexico, In-

teramericana. [1942] Technical drawing, including aeronau-tical drafting. The Macmillan Company. New York.GIMNEZ MORELL, R. V. 1988. Espacio, visin y representacinen el Dibujo y en la pintura del siglo XX. Servicio de Publi-caciones Universidad Politcnica. Valencia.

34 ANCIENT MAPS FOR THE STUDY OF THELAND DEVELOPMENT: MAPS AND DRAW-INGS FROM THE CIVIL DISPUTES INSPAIN AND THE INDIES (II)

by Pilar Chas Navarro

AbstractThe usual sources of information at a local scale ofthe American territories conquered by the SpanishCrown during the 16th and 17th centuries are the Rela-ciones Geogrficas, as well as the charts, the projects

of fortifications or public works, and the maps drawnby the jesuites and the officials of the Crown. But thereis another collection of maps that has a great impor-tance because it prov ides essential data about theland properties or the land users rights. These weremaps drawn for civil disputes concerning the own-ership or rights to particular pieces of land, which werejudged at the Royal Audiences and Chancelleries.These institutions of justice had been created in Spainin the 14th century, with similar purposes to those al-ready existing in England, France and the Low Coun-tries. They were also soon exported to the Indies withtheir own singularities.The article explores those singularities and their con-tributions to the history of the cartography and to thestudy of the evolution of the territorial frames, alongfour centuries. Such interesting cartographic mate-

rials still remain almost unknown. The collections arecomposed by more than a thousand manuscript mapsfrom the 15th to the 19th century, which show thechanging image of the Spanish and the Latinameri-can territories.

IntroductionThe searches that have focused on the Spanish ter-ritories at a local scale since the end of the 15th cen-tury, have used almost solely the usual sources of theRelaciones Topogrficas(Topographical Relations)for the Iberian peninsula, and the Relaciones Ge-ogrficas(Geographical Relations)for the Indies.Both Relacioneswere more or less extended sets ofquestions derived from an institutional initiative, thattook place mainly between 1530 and 1812 at the In-dies, and between 1575 and 1578 at the Iberian penin-sula. They were usually proposed by the Casa de Con-tratacin in Seville, with the aim of gathering severalinformation about the lands of the different provincesthat then composed the vast Spanish Empire (Vilar1970). But they also looked for reliable information inorder to get the taxes.The collectons of cadastral maps and of thosedrawn for civil disputes are also essential carto-graphic sources that are focused on describing theland ownership, the land uses and the duties relat-ed to the territories.

Documents

EGA-Aproximacion al analisis sistema diedrico español