The analytical geometry in the Curso completo de matemáticas puras (1829) by José de Odriozola

Isabel María Sánchez Sierra, María Teresa González Astudillo


While the teaching of analytical geometry in Spain at the beginning of the 19th century shows some reminiscences of the geometry of Descartes, it also incorporates the use of coordinate systems, much as is done today. The work Curso Completo de Matemáticas Puras, by José Odriozola, published
between 1827 and 1829, is a good example of this. In fact, in this book we find both ways of doing analytical geometry, although they are clearly differentiated into different sections. In this article we provide a brief biography of Odriozola, an overview of the socio-educative context of the time, and a look at the contents that were part of each of the two sections mentioned. The first section – that of analytic geometry - describes how algebraic expressions were constructed, the geometric interpretations that were given, the problems raised, the interpretation of negative solutions to these problems and the limitations of the algebraic expressions to homogeneous equations. These last two aspects were a bone of contention for every textbook author, at least during the first half of the nineteenth century. Likewise, we analyze the coordinate systems that appear in the book, along with the utility that was given to these systems for solving certain geometric problems.


Analytical geometry; Textbooks; History of education


Descartes, René. La Geometría, 1637. Buenos Aires: Espasa-Calpe, 1947. Traducido por Pedro Rossell.

Enciclopedia Universal Ilustrada Europeo-Americana. «Odriozola». Madrid: Espasa-Calpe, S.A., 1929. T. 39.

Maz, Alexander. «Los números negativos en España en los siglos XVIII y XIX». PhD diss., Universidad de Granada, 2005.

Odriozola, José. Curso Completo de Matemáticas Puras. Madrid: Imprenta que fue de García, 1827-1829.

Sánchez Sierra, Isabel M. «La Geometría Analítica en los libros de texto para secundaria y universidad en España en el siglo XIX». PhD diss., Universidad de Salamanca, 2015.

Sánchez Sierra, Isabel Mª., y Mª. Teresa González Astudillo. «La Geometría Analítico-Descriptiva de Mariano Zorraquín». HISTEMAT 2, no. 3 (2016): 200-228.

Sánchez Sierra, Isabel Mª., y Mª. Teresa González Astudillo. «La geometría analítica de Alberto Lista». Épsilon - Revista de Educación Matemática 34, no. 95 (2017): 7-24.

Sánchez Sierra, Isabel Mª., y Mª. Teresa González Astudillo. «La geometría analítica en España durante el siglo XIX: estudio de las soluciones negativas de una ecuación». Enseñanza de las ciencias 35, no. 3 (2017): 89-106.

Smith, David y Marcia Latham. The Geometry of René Descartes: Translated from de French and Latin, with a facsimil of the first edition, 1637. Illinois: Open Court, 1952.

Vea, Fernando. Las matemáticas en la enseñanza secundaria en España en el siglo XIX. Zaragoza: Seminario de Historia de la Ciencia y de la Técnica de Aragón. Facultad de Ciencias (Matemáticas), 1995.



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