Reading: Legendre: towards a rigorous and didactic geometry

In ancient Greece, axiom seemed to mean a truth common to all fields of study, and postulate, a truth of a specific field. Euclid assumed five postulates — among them, that from any point a straight line is drawn to any other, that every limited straight line extends, that with any center and radius a circle is drawn, that right angles are equal, and the famous fifth, that of parallels — and five axioms, such as “things equal to the same thing are equal to each other” and “the whole is greater than the part”. From then on, The Elements rattle off theorems without exercises or applications: a text that is not very didactic by modern standards.

Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.Rafael Sanzio · Wikimedia Commons · Public domain

In 18th-century France several books appeared to make Euclid more accessible, but many sacrificed rigor. A notable exception was Elements of Geometry by Adrien-Marie Legendre (1752–1833). Legendre shined in his research — in elliptic functions and number theory, in which he completely enunciated the law of quadratic reciprocity — and his geometry book, which reconciled rigor and didactic, had twenty editions in France during the author’s lifetime. In 1809 it was translated in Rio de Janeiro and adopted at the Royal Military Academy, created the following year. For about forty years, Legendre tried to prove the fifth postulate from the others — an impossible task, as we now know — but even in these attempts he showed great ingenuity.

Adrien-Marie Legendre, caricature by Julien-Léopold Boilly.Julien-Léopold Boilly · Wikimedia Commons · Public domain

Euler's Eyes

Reading: Legendre: towards a rigorous and didactic geometry

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