WDS'09 Proceedings of Contributed Papers, Part I, 171–175, 2009.
ISBN 978-80-7378-101-9 © MATFYZPRESS
History of Descriptive Geometry Teaching at Secondary
Schools
V. Chmelı́ková
Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.
Abstract. The aim of this article is to state some fundamental information about
the teaching of this subject in the second half of the 19th century in Bohemia. We
will have a look at the first Czech textbooks on descriptive geometry and some
fundamental information about their authors.
Introduction
Descriptive geometry as a science developed at the end of the 18th century. It was created
hand in hand with the industrial development in France (Gaspard Monge1 is known as a founder
of descriptive geometry) and consequently in other European countries. In Bohemia, descriptive
geometry was at a high level as early as in the 1840s (for example in about 1875 descriptive
geometry did not exist as a separate subject at secondary schools2 in Germany, but it was only
a part of subject drawing, which was taught two lessons per week in the last class of secondary
school).
The first lectures in Bohemia
The first lectures on descriptive geometry in Bohemia were given in 1830 at Prague Polytechnic and in 1852 at secondary schools. But these ones were given in German of course.
The first Czech lectures were initiated in 1861 by Rudolf Skuherský3 at Prague Polytechnic and
by Dominik Ryšavý4 at the first Czech realschule5 in Prague. Realschule was a special kind
of secondary school with the emphasis on mathematics and natural sciences as a contrary to
gymnasium (grammar school), where more lessons of Greek, Latin etc. were taught.
Descriptive geometry at secondary schools
In the 1880s, there were already 14 realschules, where descriptive geometry was taught in
Czech.6 The teachers were mostly graduates from Prague Polytechnic. With the beginning of
descriptive geometry teaching, publication of textbooks is connected. The first textbook on
1
Gaspard Monge, Comte de Péluse, was born on 10th May 1746 in Beaune. He became professor of mathematics at Mézières in 1768 and professor of physics in 1771 at the same place. Later he was made professor at
the École Polytechnique in Paris, where he gave lectures on descriptive geometry. These lectures were published
in 1800 in the form of a textbook entitled Géométrie descriptive. He died on 28th July 1818 in Paris.
2
In this article, I use the term secondary schools in meaning to schools for students between the ages approximately from 11 to 18 years.
3
Rudolf Skuherský was born on 23th April 1828 in Opočno. He studied at Prague Polytechnic and in 1854
he became the first professor of descriptive geometry at this school. Skuherský died because of diphtheria on
9th October 1863 in Prague. More see [Šišma, 2003].
4
Dominik Ryšavý was born on 12th October 1830 in Bı́tovany. He worked as a secondary school professor at
a realschule in Prague and is known as the author of the first Czech textbook on descriptive geometry Zobrazujı́cı́
měřictvı́. He died on 26th September 1890 in Prague.
5
For this type of school the English term does not exist, therefore in this article I use the term realschule
(realschules – pl.) from German – die Realschule (in Czech – reálka). Lower realschules were for students from
11 to 15 years, higher realschules were from students from 15 to 18 years. Age intervals are only approximate.
6
The list of schools where descriptive geometry was taught in Czech according to [Lavička, 1883]: Karlı́n
(Prague), Nové Město (Prague), Hradec Králové, Kutná Hora, Litomyšl, Pardubice, Pı́sek, Plzeň, Rakovnı́k,
Tábor, Brno, Prostějov, Přerov, Telč.
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CHMELÍKOVÁ: HISTORY OF DESCRIPTIVE GEOMETRY
descriptive geometry written in Czech is the textbook Zobrazujı́cı́ měřictvı́ [Ryšavý, 1862–3]
(zobrazujı́cı́ měřictvı́ is the former Czech term for descriptive geometry) by Dominik Ryšavý.
Its first part was published in 1862, the second part in 1863. It contained many mistakes, but
in spite of them was very important for Czech schools. Also, the Czech terminology had to be
created at that time and its uniformity was hardly achieved.
Figure 1. The title pages of the first and second part of Ryšavý’s textbook
Another textbook – Deskriptivnı́ geometrie pro vyššı́ třı́dy škol střednı́ch [Descriptive Geometry for Higher Classes of Secondary Schools] [Šanda, 1877] by František Šanda7 – was published in 1877. However, it is not much better than the former one, because there are similar
mistakes. For example, in Ryšavý’s textbook, there is the figure of two intersecting cylinders
on the page 90 (in the second part), but the intersection is drawn incorrectly. The same figure
can be found in the textbook by Šanda on the page 215.
In my opinion, the best descriptive geometry textbooks of that time were the textbooks by
Vincenc Jarolı́mek,8 about which I write below.
7
František Šanda was born on 27th December 1831 in Nové Město near Chlumec nad Cidlinou. He worked as
a teacher of drawing, mathematics and descriptive geometry at secondary schools in Prague, Košice (Slovakia)
and Tábor. Šanda died on 15th November 1893 in Prague. More see [Šišma, 2003]
8
Vincenc (somewhere you can read Čeněk, but it’s only the older form of this name) Jarolı́mek was born on
25th June in Pardubice. His father (also Vincenc or Čeněk Jarolı́mek) was a teacher of maths and author of
mathematics textbooks for lower realschule. Vincenc junior studied at the lower realschule in Pardubice, higher
realschule in Kutná Hora, Pı́sek and Prague and then at the Prague Polytechnic. After his studies he worked
one year in the Ringhoffer’s engineering factory, but he wanted to be a teacher, therefore he became assistant of
the professor Josef Webr at a realschule in Prague. Later he was professor at a realschule in Prague, director of
realschules in Hradec Králové, Pı́sek and Prague and provincial school inspector. At the same time he stood in
lectures on descriptive geometry in Brno for professor Jan Sobotka, called up to Charles University in Prague.
On 25th January 1907 he became regular professor of descriptive geometry at Prague Polytechnic. Vincenc
Jarolı́mek was a member of many institutions, let me also mention that he was a member of the Union of Czech
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CHMELÍKOVÁ: HISTORY OF DESCRIPTIVE GEOMETRY
I have found the curriculum for realschules in accordance with the order of 23th June 1880.
Compared to recent teaching of descriptive geometry at secondary schools, there was in addition
included lighting, rotating around axis, fourth plane of projection, intersection of polyhedra,
curved surfaces, intersection of curved surfaces, central projection and other topics.
Descriptive geometry was taught three lessons per week from the fifth to the last seventh
class. Beside teaching at schools, the homework was important too. The school-leaving examination in descriptive geometry lasted five hours. Here are some examples of the schoolleaving examination exercises which I have found in the annual general reports of Czech Prague
realschule.9 These exercises were used in the school year 1895/1896 which was the 47th year
from establishment of this school and Vincenc Jarolı́mek was the director of the school at that
time.
First exercise
There are given the lines A and B. Construct a line C which is parallel with the axis x so
that the lines A, C and the lines B, C are the concurrent lines. Moreover find an angle between
the line C and a plane ̺ in the case that the plane ̺ is parallel with the lines A and B.
A[a(−5, 2, 0), b(6, 9, 10)], B[c(−3, 7, 3.5), d(0, 4, 10)].
Second exercise
There are given an ellipsoid of revolution and a cylinder of revolution. Construct an intersection of the ellipsoid with the cylinder. Moreover construct a tangent in arbitrary point of
intersection and develop the part of cylinder which is inside of ellipsoid.
Ellipsoid of revolution: vertices a(−1, 6, 11), b(−1, 6, 1); radius of equator 4.
Cylinder of revolution: axis c(−2.5, 6, 11), d(3.5, 6, 1); radius 3.
Third exercise
There are given a regular quadrilateral pyramid abcdv, its bottom abcd is in the first plane of
projection, and an orthogonal trilateral prism ef ghij with a bottom ef g. Construct intersection
of these solids in central projection and their lighting.
Regular quadrilateral pyramid: a(−8, 0, 0), c(−14, −10, 0); high 14.
Orthogonal trilateral prism: e(−5.75, −0.5, 0), f (−4.25, −6.5, 0), g(−5, −3.5, 7),
h(−17, −6.5, 7).
Centre of projection s(0, 12, 10), vanishing point of lighting rays us1 (9, 0, −2).
As you can see, school-leaving examination exercises were very difficult and today’s students
would have certainly problems with these constructions, because many topics of descriptive
geometry, which are contained in these exercises, are not taught at secondary school currently.10
Mathematicians (later this organisation was called the Union of Czech Mathematicians and Physicists, in Czech
JČMF – Jednota českých matematiků a fyziků). Jarolı́mek retired in 1915 and died of stroke suddenly on 14th
December 1921 in Prague when he went from a dentist. More see [Kounovský, 1946].
9
At that time, points were usually denoted with small (lower case) letters and lines with capital (upper case)
letters.
10
In the second exercise students have to know ellipsoid of revolution and its intersection with cylinder, in the
third example they need central projection and lighting. These parts of descriptive geometry are usually not
taught at present.
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CHMELÍKOVÁ: HISTORY OF DESCRIPTIVE GEOMETRY
Figure 2. Vincenc Jarolı́mek with his signature
The textbooks by Vincenc Jarolı́mek
Apart from textbooks on mathematics, especially on geometry,11 and a textbook for high
technical schools Deskriptivnı́ geometrie pro vysoké školy technické [Descriptive Geometry for
Technical Universities] (1909), which he wrote together with Bedřich Procházka,12 Vincenc
Jarolı́mek wrote the collection of exercises Deskriptivnı́ geometrie v úlohách13 [Descriptive
Geometry in Exercises] [Jarolı́mek, 1873] and the textbook in three parts Deskriptivnı́ geometrie
pro vyššı́ školy reálné [Descriptive Geometry for Higher Realschules] [Jarolı́mek, 1875–7].14
The greatest advantage of these textbooks as compared to the former ones are systematical
approach and accurate and logical explanations. Jarolı́mek always begins with the explanations
of basic terms and then proceeds towards more difficult ones. His textbooks were so good
that there was a high demand for them even abroad. For example Deskriptivnı́ geometrie pro
11
The on geometry by Vincenc Jarolı́mek for realschules: Geometrie pro 4. třı́du škol reálných [Geometry for
4 Class of Realschules] (1874), Nauka o tvarech měřických pro 1. třı́du reálných škol [Teaching About Measuring
Shapes for 1st Class of Realschules] (1890) and Geometrie pro 2. a 3. třı́du škol reálných [Geometry for 2nd and
3rd Class of Realschules] (1891).
th
12
Bedřich Procházka was born on 4th July 1855 in Rakovnı́k. He studied at the Prague Polytechnic and then
he worked as an assistant of descriptive geometry at the Polytechnics in Germany and Prague. From 1887 he
taught at secondary schools (Chrudim, Pardubice, Prague–Karlı́n, Náchod – here he became director of school).
In 1904 he became professor of descriptive geometry at the Czech Polytechnic in Brno, from 1908 he was rector
of the Prague Polytechnic. Procházka retired in 1925 and died on 3th January 1934 in Prague. More see [Šišma,
2003].
13
Deskriptivnı́ geometrie v úlohách [Descriptive Geometry in Exercises] was firstly published in 1873 and the
second edition of this collection in 1880. This book in the first edition includes 1000 exercises which are sorted
according to the topic of descriptive geometry and moreover there are instructions on how to draw some difficult
exercises. The second edition was modified. It includes 1224 exercises, but without instructions.
14
The first part of this textbook was firstly published in 1875, the second part in 1876 and the third part
in 1877. The second edition (1887) of Deskriptivnı́ geometrie pro vyššı́ školy reálné [Descriptive Geometry for
Higher Realschules] was abridged (the first edition – more than 400 pages, other editions – less than 300 pages),
in other editions (this book was published five times in total) there were only little changes.
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CHMELÍKOVÁ: HISTORY OF DESCRIPTIVE GEOMETRY
vyššı́ školy reálné [Descriptive Geometry for Higher Realschules] was translated into Bulgarian
and was published in Plovdiv in 1895. Jarolı́mek’s textbooks were used until 1909, then they
were replaced by new textbooks by Josef Pithardt and Ladislav Seifert, because in 1908 a new
curriculum was created. Contribution of textbooks by Vincenc Jarolı́mek was analyzed in
articles by Jaroslav Folta [Folta, 1958, 1959].
Conclusion
This article covers only the history of descriptive geometry teaching in Bohemia in the second half of the 19th century. I tried to give a short overview of the development of descriptive
geometry teaching and to introduce the main persons who were responsible for this development.
Acknowledgments.
help with the dissertation.
The author thanks doc. RNDr. Jindřich Bečvář, CSc., for his professional
References
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