Table Of ContentInternational Studies in the History of Mathematics
and its Teaching
Series Editors: Alexander Karp · Gert Schubring
Évelyne Barbin
Marta Menghini
Klaus Volkert Editors
Descriptive
Geometry, The
Spread of a
Polytechnic Art
The Legacy of Gaspard Monge
International Studies in the History
of Mathematics and its Teaching
SeriesEditors
AlexanderKarp
TeachersCollege,ColumbiaUniversity,NewYork,NY,USA
GertSchubring
UniversitätBielefeld,Bielefeld,Germany
UniversidadeFederaldoRiodeJaneiro,RiodeJaneiro,Brazil
The International Studies in the History of Mathematics and its Teaching Series
creates a platform for international collaboration in the exploration of the social
history of mathematics education and its connections with the development of
mathematics. The series offers broad perspectives on mathematics research and
education, including contributions relating to the history of mathematics and
mathematics education at all levels of study, school education, college education,
mathematicsteachereducation,thedevelopmentofresearchmathematics,therole
of mathematicians in mathematics education, mathematics teachers’ associations
andperiodicals.
The series seeks to inform mathematics educators, mathematicians, and historians
aboutthepolitical,social,andculturalconstraintsandachievementsthatinfluenced
thedevelopmentofmathematicsandmathematicseducation.Insodoing,itaimsto
overcomedisconnectednationalculturalandsocialhistoriesandestablishcommon
cross-cultural themes within the development of mathematics and mathematics
instruction.However,atthecoreofthesevariousperspectives,thequestionofhow
tobestimprovemathematics teachingandlearningalwaysremainsthefocalissue
informingtheseries.
Moreinformationaboutthisseriesathttp://www.springer.com/series/15781
Évelyne Barbin (cid:129) Marta Menghini (cid:129) Klaus Volkert
Editors
Descriptive Geometry,
The Spread of a Polytechnic
Art
The Legacy of Gaspard Monge
123
Editors
ÉvelyneBarbin MartaMenghini
UFRSciencesetTechniques DepartmentofMathematics
Nantes,France Sapienza–UniversityofRome
Rome,Italy
KlausVolkert
UniversityofWuppertal
Wuppertal,Germany
ISSN2524-8022 ISSN2524-8030 (electronic)
InternationalStudiesintheHistoryofMathematicsanditsTeaching
ISBN978-3-030-14807-2 ISBN978-3-030-14808-9 (eBook)
https://doi.org/10.1007/978-3-030-14808-9
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Preface
The Purposeofthe Book
The developments that followed the French Revolution of 1789 led to the birth of
newkindsofschoolsandanewkindofmathematicalteaching.In1794–1795,two
entirelynovelinstitutionsofhighereducationwerefoundedinFrance.Oneofthem
wastheÉcolenormaledel’anIII,whichwasdevotedtomultistepteachereducation
for the new schools of the revolution. A second was the École polytechnique
(initiallyÉcoledestravauxpublics),whichprovidedtrainingforcivilandmilitary
engineers and would serve as a model for many countries in the future.1 One of
the leading actors of the period was Gaspard Monge, whose lessons delivered to
futureteachersattheÉcolenormaleweresummarizedinhiswell-knowntextbook
Géométriedescriptive(1799).Mongewasalsoinfluentialinelaboratingtheinitial
conceptionoftheÉcolepolytechniqueandplayedaconsiderableroleinit.Thecase
of descriptive geometry is highly noteworthy: being taught publicly the first time
at two institutions created in the revolutionary period, later it exemplified a novel
impactofmathematicsonsociety.
The military and civilian needs for well-prepared officers, engineers and ad-
ministrators, which led to the creation of the École polytechnique and later of the
École Centrale des Arts et Manufactures, were common to many countries. New
typesofschoolswerecreatedeverywherelikePolytechnische SchuleninGerman-
speaking countries, Politecnici in Italy and military schools in the USA. They all
aimed to address the needs of their students in scientific and technical studies.
Descriptive geometry was an important subject in these new schools. New ways
of understanding and teaching it were created in different places in response to
the needs and teaching traditions of different countries and regions. For instance,
in France—and in other countries—descriptive geometry was soon considered to
beanewtypeofgeometryandwastaughttoallstudentsinthesecondpartofthe
1MoreontheminChap.1.
v
vi Preface
century.Inotherplaces,itsinterestremainedrestrictedtoitstechnicalnature,andits
disseminationwasconnectedwiththecreationofmilitaryacademiesandanalogous
institutions.
Thisbookintendstoanalysehowdescriptivegeometryspreadthroughgeneral,
technical or artistic education and thus in “polytechnic” schools, in different
countries around the world. Our purpose is to go beyond the analysis of the
development of the subject in individual countries. We do so by gathering and
confronting the local experiences in order to highlight the parallel processes and
theconnectionsamongthem.Wealsoseektounderstandhowtheymayhavebeen
providedbythecirculationbetweencountriesandpersonsorbypublicationsrelated
tooursubject.
Depending on different regions, the period that is considered here ranges from
approximately 1800 up to the 1900s and beyond. The titles of the contributions
oftenrefertoacountryoraregion.Ofcourse,itisdifficulttospeakofcountriesin
thenineteenthcenturyinrelationtoGerman-speakingregionsandalsoinrelationto
Italybeforeitsunification—nottospeakofthecomplicatedsituationintheBalkans.
Similarly,thereisacontributionaboutteachingintheCzechlanguagealthoughthe
country wasn’t created until 1918. Moreover, the considered period corresponds
to a political and colonial situation that influenced relations among countries (for
instance, those between France, Greece and Italy at the beginning of the century
andlaterbetweenPortugalandBrazil),whichmakesattributingthedissemination
processtoonecountryevenmoredifficult.
The process of the introduction of descriptive geometry is studied at different
levels: the chapters may be devoted to the countries but also to specific towns
and regions. The Austro-Hungarian Imperium is represented mainly by Vienna
(Chaps.11and12)andpartlybyPragueandBrno(Chap.16)aftertheintroduction
oftheCzechlanguage.Russiaisrepresentedbyachiefcharacter,CharlesPoitierin
Saint Petersburg (Chap.13). Latin America (Chap.21) deals mainly with Brazil—
wherehistoricalresearchonthesubjectisthemostdeveloped.Sometimes(although
rarely),differentchaptersconsiderthesameobjects—itseemsvaluablebecauseof
thedifferentcontext.
Notallcountriesinvolvedintheprocessareincludedinthisbook:othercountries
likeCanada,Sweden,BelgiumandtheFarEastcouldbeconsidered.Nevertheless,
the 22 chapters of this book, written by different authors from 15 countries, offer
awidepanoramaandcanalsohelptoengagestudiesinthefieldconcerningother
countriesorregions.
The content of the book is discussed below, but here, we need to make a
note on some chapters because they address some special aims. The chapter on
the Vienna School gives an account of the lives of the involved characters, of
the academic life in this institution and of the work of the students (Chap.12).
In this way, the chapter addresses an aim to explore the academic world from
another “human” perspective. Simultaneously, we aimed to take a closer look
at the relations between projective and descriptive geometry. To rise descriptive
geometry to a level of an exact science was, as the reader will see, a concern
in many countries; in the work of Wilhelm Fiedler (Chap.10) and also of Luigi
Preface vii
Cremona(Chap.4),projectivegeometrywasconsideredtobethetheoreticalbasis
fordescriptivegeometry—theconnectionbetweenthetwoareasbeingestablished
through the concept of central projection. Fiedler’s and Cremona’s point of view
isplacedinthecontextof“moderngeometry”inthesecondpartofthenineteenth
century even if—in their case—attention was paid mainly to synthetic geometry.
This “purist” trend influenced the development not only of descriptive geometry
but of all geometry in some universities (e.g.Barcelona and Madrid, as shown in
Chap.6). The connection between projective and descriptive geometry epitomized
in some way the relationship between pure and applied sciences and became also
central in the relationship between the faculties of sciences and the schools for
engineersandarchitects(Chaps.4and6).
The Content
The chapters of the book are given in a geographical order, which begins from
the “near” countries of Southern Europe (which also includes Greece) and moves
to the more culturally independent countries of Central and Eastern Europe and
thentoEnglandandlastlytonon-Europeancountrieseffectedbycolonizationand
decolonization(theAmericasbutalsoEgypt).Thegeographicalordercanbroadly
correspondtoachronologicalorderreferringtotheintroductionoftheteachingof
descriptivegeometry.
TheInitialSpreadofDescriptiveGeometryinSouthernEurope Thefirstpart
of the book concerns countries in Southern Europe: France with Italy and Spain,
PortugalandGreece;fourcountriesstronglyinfluencedbyFranceatthebeginning
of the nineteenth century before and during the Napoleonic wars. This influence
resulted in the creation of new schools for engineers and military officers inspired
not only by the École polytechnique but by French schools in general and in the
importance given to French mathematics, especially to descriptive geometry. The
textbook of Gaspard Monge (Chap.1) was translated into Spanish, Italian and
Portuguesebetween1803and1812.Locally,peopleconnecteddirectlywithFrance
playedanimportantrolelikeAgustindeBetancourtinSpain(Chap.5)andIoannis
PapadakisinGreece(Chap.8).Thechaptersconcerningthesefivecountriesindicate
thatinteractionsbetweenthemcontinuedafterthisfirstperiod.Indeed,theteaching
of descriptive geometry quickly spread to new kinds of schools and was taught in
universitiessuchasinItalyandinPortugal(Chaps.3and7).Duringthisperiod,the
situationinFrancechangedasdescriptivegeometrybecameasubjectofpreparatory
gradesforpublicschools(Chap.2).Inallthesoutherncountries,thedissemination
of descriptive geometry was followed by a lasting establishment of the subject in
thecurricula,inthecourseofwhichmanybooksbyFrenchauthorsweretranslated.
Translations from French continued after the middle of the century when
new conceptions concerning the teaching of descriptive geometry attracted new
audiences, especially those related to secondary teaching. Such is the case of the
viii Preface
textbooksbyAntoineAmiot,translatedintoItalianin1875(Chap.3),andbyPaul-
LouisCirodde,translatedintoSpanishin1865(Chap.5).TranslationsfromFrench
are more rare in Portugal, where authors wrote new textbooks with the help of
different foreign textbooks, and in Greece, where it was not necessary to translate
becausestudentshadtolearnFrench.
In the last quarter of the century, the political context changed considerably;
however, circulation between the five countries continued with the teaching of
projectivegeometryandtheimportancegiventoschoolsforengineersorforpeople
involved in other technical professions. Then, translations from languages other
thanFrenchemerged,particularlywiththetextbookbyLuigiCremona,whichwas
translatedfromItalianintoFrenchtwoyearsafteritsoriginalpublicationin1875,
into German in 1882 and into English in 1885. In Southern Europe countries—as
wellasinotherEuropeancountries—descriptivegeometryatsecondaryeducation
levelcameclosertotheteachingofprojectivegeometry,anditwasfinallyincluded
in projective geometry at the turn of the century. But it remained an independent
subject for students in the arts and architecture (Chap.4) and for candidates of
GrandesécolesinFranceuntiltheyear1970(Chap.2).
The Establishment of Descriptive Geometry in Central and Eastern Europe
LikeinFrance,therehadexistedsometraditionofteachingintechnicalschoolsin
Central and Eastern Europe, although in general at a low level. Together with the
dissemination of descriptive geometry, these schools were upgraded in their level
andintheirdisciplinaryrange.Forinstance,inGerman-speakingcountries,different
kindsoftechnicalsecondaryschoolsexisted,whichthenevolvedintopolytechnics.
Different kinds of Realanstalten could be, on the other hand, considered as the
predecessors of the Technische Hochschulen (Chap.9). The situation in German-
speakingcountrieswasindeedcomplicated:evenafterthecreationoftheso-called
secondGermanempire,thestateswereautonomousconcerningteaching.
InGermanyandinCentralEurope,itwasquiteusualtofindmethodsindescrip-
tivegeometrydifferentfromMonge’sprojectionsandatendencydevelopedtowards
using the methods of projective geometry in descriptive geometry (Chap.10); we
alsofind—insomeothercases—aseriesofmodifications toavoidperceived diffi-
cultieswithrespecttotheFrenchtradition,inparticularwhenteachingtoarchitects
and engineers. In Austria, the emphasis was initially on empirical methods, and,
therefore,thegeometricmethodsneededforarchitectsandengineersweretaughtin
coursessuchasengineeringorarchitecturaldrawing.Later,aneedforprogressin
scientificandtechnicalfieldsbroughtanautonomousexpansionoftheteachingof
descriptive geometry at the university level (Chap.11). Descriptive geometry was
oneofthemainsubjectsalsoinCzechuniversities,butMonge’sprojectionmethod
wasnotincluded,and,instead,wefindtheoriesofcurvesandsurfaces,collineations
and projective and kinematic geometry; this was rather the Viennese influence. A
newmethodfororthogonalprojectionwaselaboratedbySkuherský(Chap.16).In
Czech (real) secondary schools in the 1870s, the descriptive geometry exam was
one of the obligatory parts of the examination system and was very difficult in
comparisontoothercontemporaryrequirementsinsecondaryeducation.
Preface ix
New colleges inspired by the École polytechnique appeared in Denmark in the
1830s. The Danish Polytechnic College deserves particular attention for its high
academic ideals. Hielmlev’s geometry of reality raised descriptive geometry to
a new and higher scientific level (Chap.15). Indeed, this approach to geometry
focusedontheaccuracywithwhichonecanexecutegeometricalconstructionsina
systematic(theoretical)way.IntheNetherlands,thelackofwell-trainedengineers
during the eighteenth century was seen as the cause of different problems in the
country: mainly the lack of military success but also the management of, or lack
of, flood controls. Political changes brought new laws in education leading to the
renewal of primary education, the creation of a sort of “modern” curriculum for
secondary school and the creation of schools for the training of engineers and of
military schools. All these schools had descriptive geometry in their programmes.
Descriptive geometry—also seen as a means to facilitate the study of analytic
geometryandtotrainintellectandimagination(Chap.14)—becamepartofthefinal
examinationsinsecondaryschoolsheretoo(HBS).
InterestingistherolethattheteachingofdescriptivegeometryhadinSerbia;it
startedinthe1850sandlastedtillthetwenty-firstcentury.Thislongevitywasdue
toaninterestinpracticalmathematicsandtotheabsenceofaEuclideantradition.
Politically,SerbiawasinfluencedbytheHabsburgsmorethanbytheOttomansand
mathematically by the geometry of Fiedler, and descriptive geometry was seen as
animportantmethodtostudygeometryandmathematics(Chap.17).Incontrastto
Serbia, the existence of a “local” geometric tradition in England was a hindrance
tothedevelopmentofdescriptivegeometry.DuealsotothewarsbetweenEngland
and France, the translation of texts on descriptive geometry and its introduction
into the educational system happened only after 1840s. Moreover, in this period,
England was an important contributor to applied mathematics, and its interest in
practicalgeometrybroughttolifeapproacheslikeFarish’s“isometricperspective”.
Itisinterestingtonotethatpolytechnicschoolsdidnotfullydevelopthereuntilthe
twentiethcenturyalthoughthefirstsuchschoolwasfoundedin1838(Chap.18).
OneoftherarecasesofadirecttransmissionofdescriptivegeometryfromFrance
to the continental countries is represented by Russia; indeed, one of its leading
characters, Charles Poitier, was invited to Russia as a former student of the École
polytechnique and started there the teaching of descriptive geometry. Although
closely aligned with Monge, in his works, he enriched the theory of surfaces and
theirstudyandapplications,particularlyinprojectsrelatedtoshipbuildingandthe
buildingofmachines(Chap.13).
Descriptive Geometry in America and Africa A very interesting feature of
descriptive geometry is the fact that it soon reached many faraway regions. In the
USA,theteachingofdescriptivegeometrybecameestablishedrelativelyslowlyand
hadamajorlinktothatofgeometryingeneralandwasevenseenasitscontinuation.
Its teaching was influenced by France at the beginning of the nineteenth century.
Authorsgenerallyintroducedthemethodofprojectionsinasimplifiedwaytoadapt
it to the new readership until it established its place in emerging institutions as a
graphicartforthetrainingofengineers(Chap.19).InLatinAmerica,withparticular
