The analytical geometry in the Curso completo de matemáticas puras (1829) by José de Odriozola - Sánchez Sierra, Isabel María; González Astudillo, María Teresa (Historia y Memoria de la Educación, 2019)
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.