• In 1585 Giovanni Battista Benedetti (1530–1590) published the first general constructions of ovals given the axes. • They were included in a letter to Pietro Pizzamano (1512–1571). • It is plausible that Benedetti discovered them when solving a purely geometrical problem posed by Pizzamano, described in a separate letter. • Only one of the two constructions was included in recent surveys on oval constructions but wrongly attributed to much later authors. Giovanni Battista Benedetti (1530–1590) derived two constructions of ovals given their minor and major axes. These were published in 1585 and seem to be the first solution to this problem. Therefore, the generally accepted view that “the geometrical construction for drawing an oval for any given proportion was not known in the sixteenth century” (Ana López Mozo, 2011; Nexus Network Journal, p. 571) has to be revised. We provide a transcription of Benedetti’s Latin text and a translation, analyze his mathematical reasoning in detail, and review Benedetti’s impact on the construction of ovals.

This paper aims to shed light on Bolzano’s position on language and meaning as part of the “Sprachphilosophie” current in 19th century Germany. Studied most often from the Wissenschaftslehre , this position is already at work in the early mathematical writings, e.g. in the Rein analytischer Beweis . Bolzano shows the need to develop objective semantics as a prerequisite for the deductive rigor of proofs. We must first fix the meaning of words denoting concepts, and carefully formulate propositions in order to specify their mathematical content and refer it to the corresponding discipline. This is how the autonomy of Analysis is established not only in relation to Geometry but also in relation to Algebra. Cet article vise à mettre en lumière la position de Bolzano sur le langage et la signification, qui s’inscrit dans le courant de « Sprachphilosophie » dans l’Allemagne du XIX e siècle. Étudiée le plus souvent à partir de la Wissenschaftslehre , cette position est déjà à l’œuvre dans les premiers écrits mathématiques, par exemple dans le Rein analytischer Beweis . Bolzano y montre la nécessité d’établir une sémantique objective comme préalable à la rigueur déductive des preuves. Il faut commencer par fixer le sens des mots désignant les concepts, formuler avec soin les propositions, spécifier leur contenu mathématique et le référer à la discipline correspondante. L’autonomie de l’Analyse est ainsi établie non seulement par rapport à la Géométrie, mais aussi par rapport à l’Algèbre.