Benedetti's constructions of ovals

TL;DR

Giovanni Benedetti's early 16th-century constructions for drawing ovals with given axes were pioneering, challenging the belief that such geometric solutions were unknown before the 17th century.

Contribution

This paper provides the first translation and detailed analysis of Benedetti's original methods for constructing ovals, highlighting their historical significance.

Findings

Benedetti's constructions predate previous known solutions

His methods were rarely acknowledged in later literature

The paper offers the first English translation of Benedetti's work

Abstract

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 L\'opez-Mozo, 2011; Nexus Network Journal, p. 571) has to be revised. We provide a transcription of Benedetti's Latin text as well as a translation into English and analyze his mathematical reasoning in detail. This is the first main focus of the present note. The second focus is the impact of Benedetti's contributions to ovals. In particular, it is observed that they were rarely mentioned in the mathematical and architectural literature.