Fray Juan de Ortega's approximations, 500 years after

TL;DR

This paper explores the historical approximations of square roots in Fray Juan de Ortega's 1512 arithmetic book, revealing how these approximations were derived using methods consistent with early 16th-century mathematics.

Contribution

It uncovers the method used by Ortega to obtain his approximations, which had been previously unknown, and demonstrates their consistency with contemporary mathematical knowledge.

Findings

Twelve approximations verify Pell's equation, indicating optimality.

The method aligns with mathematical practices of the early 16th century.

Historical approximations can be reconstructed using modern understanding.

Abstract

In 1512, on December 30, the first edition of Fray Juan de Ortega's Arithmetic was published in Lyon. The last chapter, titled "Rules of Geometry", deals with lower approximations of 14 square roots. In later editions of the Arithmetic on 1534, 1537 and 1542 in Seville, these values are replaced by upper approximations. Twelve of them verify the Pell's equation, and so they are optimal. At this moment nobody knows the way they were obtained. In this paper we show how these approximations can be obtained through a method consistent with the mathematical knowledge at that time.